0,3

Also number of nonnegative solutions to b + 2c + 3d + 4e + ... = n and the number of nonnegative solutions to 2c + 3d + 4e + ... <= n. - Henry Bottomley, Apr 17 2001

a(n) is also the number of conjugacy classes in the symmetric group S_n (and the number of irreducible representations of S_n).

Also the number of rooted trees with n+1 nodes and height at most 2.

Coincides with the sequence of numbers of nilpotent conjugacy classes in the Lie algebras gl(n). A006950, A015128 and this sequence together cover the nilpotent conjugacy classes in the classical A,B,C,D series of Lie algebras. - Alexander Elashvili, Sep 08 2003

Number of distinct Abelian groups of order p^n, where p is prime (the number is independent of p). - Lekraj Beedassy, Oct 16 2004

Number of graphs on n vertices that do not contain P3 as an induced subgraph. - Washington Bomfim, May 10 2005

Numbers of terms to be added when expanding the n-th derivative of 1/f(x). - Thomas Baruchel, Nov 07 2005

Sequence agrees with expansion of Molien series for symmetric group S_n up to the term in x^n. - Maurice D. Craig (towenaar(AT)optusnet.com.au), Oct 30 2006

Also the number of nonnegative integer solutions to x_1 + x_2 + x_3 + ... + x_n = n such that n >= x_1 >= x_2 >= x_3 >= ... >= x_n >= 0, because by letting y_k = x_k - x_(k+1) >= 0 (where 0 < k < n) we get y_1 + 2y_2 + 3y_3 + ... + (n-1)y_(n-1) + nx_n = n. - Werner Grundlingh (wgrundlingh(AT)gmail.com), Mar 14 2007

Let P(z) := Sum_{j>=0} b_j z^j, b_0 != 0. Then 1/P(z) = Sum_{j>=0} c_j z^j, where the c_j must be computed from the infinite triangular system b_0 c_0 = 1, b_0 c_1 + b_1 c_0 = 0 and so on (Cauchy products of the coefficients set to zero). The n-th partition number arises as the number of terms in the numerator of the expression for c_n: The coefficient c_n of the inverted power series is a fraction with b_0^(n+1) in the denominator and in its numerator having a(n) products of n coefficients b_i each. The partitions may be read off from the indices of the b_i. - Peter C. Heinig (algorithms(AT)gmx.de), Apr 09 2007

A026820(a(n),n) = A134737(n) for n > 0. - Reinhard Zumkeller, Nov 07 2007

Equals row sums of triangle A137683. - Gary W. Adamson, Feb 05 2008

a(n) is the number of different ways to run up a staircase with n steps, taking steps of sizes 1, 2, 3, ... and r (r <= n), where the order is not important and there is no restriction on the number or the size of each step taken. - Mohammad K. Azarian, May 21 2008

Equals the eigenvector of triangle A145006 and row sums of the eigentriangle of the partition numbers, A145007. - Gary W. Adamson, Sep 28 2008

Starting with offset 1 = INVERT transform of (1, 1, 0, 0, -1, 0, -1, ...), where A080995, the characteristic function of A001318 (1, 2, 5, 7, 12, ...) is signed (++ -- ++, ...) as to 1's. This is equivalent to lim_{n>=1} A145006^n as a vector. The INVERT transform of (1, 1, 0, 0, -1, ...) begins (1, 2,..) then for each successive operation we take a dot product of (1, 1, 0, 0, -1, ...) in reverse and the ongoing results of our series (1, 2, 3, 5, 7, ...) then add the result to the next term in (1, 1, 0, 0, -1, ...). For example, a (7) = 15 = (0, -1, 0, 0, 1, 1) dot (1, 2, 3, 5, 7, 11) = (0*1, (-1)*2, 0*3, 0*5, 1*7, 1*11) = (-2 + 7 + 11) = 16, then add to (-1) = 15. - Gary W. Adamson, Oct 05 2008

Convolved with A147843 = A000203 prefaced with a zero: (0, 1, 3, 4, 7, ...). - Gary W. Adamson, Nov 15 2008

From Gary W. Adamson, Jun 12 2009: (Start)

Equals an infinite convolution product_(1, 1, 1, ...)*(1, 0, 1, 0, 1, ...)*(1, 0, 0, 1, 0, 0, 1, ...)*(1, 0, 0, 0, 1, 0, 0, 0, 1, ...)*...; = a*b*c*...; where a = (1/(1-x)), b = (1/(1-x^2)), c = (1/(1-x^3)), etc. An array by rows: row 1 = a, row 2 = a*b, row 3 = a*b*c, ...; gives:

   1, 1, 1, 1, 1, 1,  1,  1,  1,  1, ... = (a)

   1, 1, 2, 2, 3, 3,  4,  4,  5,  5, ... = (a*b)

   1, 1, 2, 3, 4, 5,  7,  8, 10, 11, ... = (a*b*c)

   1, 1, 2, 3, 4, 5,  6,  9, 11, 17, ... = (a*b*c*d)

   1, 1, 2, 3, 5, 5,  7, 10, 13, 18, ... = (a*b*c*d*e)

   1, 1, 2, 3, 5, 7, 11, 14, 20, 25, ... = (a*b*c*d*e*f)

   1, 1, 2, 3, 5, 7, 11, 15, 21, 27, ... = (a*b*c*d*e*f*g)

   1, 1, 2, 3, 5, 7, 11, 15, 22, 28, ... = (a*b*c*d*e*f*g*h)

   1, 1, 2, 3, 5, 7, 11, 15, 22, 29, ... = (a*b*c*d*e*f*g*h*i)

  ... with rows tending to A000041. Partition triangles A058398 = ascending antidiagonals. Partition triangle A008284 reversal of A058398. (End)

Starting with offset 1 = row sums of triangle A168532. - Gary W. Adamson, Nov 28 2009

P(x) = A(x)/A(x^2) with P(x) = (1 + x + 2x^2 + 3x^3 + 5x^4 + 7x^5 + ...),

  and A(x) = (1 + x + 3x^2 + 4x^3 + 10x^4 + 13x^5 + ...),

  and A(x^2) = (1 + x^2 + 3x^4 + 4x^6 + 10x^8 + ...), where A092119 = (1, 1, 3, 4, 10, ...) = Euler transform of the ruler sequence, A001511. - Gary W. Adamson, Feb 11 2010

Equals row sums of triangle A173304. - Gary W. Adamson, Feb 15 2010

p(x) = A(x)*A(x^2), A(x) = A174065; p(x) = B(x)*B(x^3), B(x) = A174068. Equals row sums of triangles A174066 and A174067. - Gary W. Adamson, Mar 06 2010

Triangle A113685 is equivalent to p(x) = p(x^2) * A000009(x). Triangle A176202 is equivalent to p(x) = p(x^3) * A000726(x). - Gary W. Adamson, Apr 11 2010

A sequence of positive integers p = p_1 ... p_k is a descending partition of the positive integer n if p_1 + ... + p_k = n and p_1 >= ... >= p_k. If formally needed p_j = 0 is appended to p for j > k. Let P_n denote the set of these partition for some n >= 1. Then a(n) = 1 + Sum_{p in P_n} floor((p_1-1)/(p_2+1)). (Cf. A000065, where the formula reduces to the sum.) Proof in Kelleher and O'Sullivan (2009). For example a(6) = 1 + 0 + 0 + 0 + 0 + 1 + 0 + 0 + 1 + 1 + 2 + 5 = 11. - Peter Luschny, Oct 24 2010

Let n = Sum( k_(p_m) p_m ) = k_1 + 2k_2 + 5k_5 + 7k_7 + ..., where p_m is the m-th generalized pentagonal number (A001318). Then a(n) is the sum over all such pentagonal partitions of n of (-1)^(k_5+k_7 + k_22 + ...) ( k_1 + k_2 + k_5 + ...)! /( k_1! k_2! k_5! ...), where the exponent of (-1) is the sum of all the k's corresponding to even-indexed GPN's. - Jerome Malenfant, Feb 14 2011

From Jerome Malenfant, Feb 14 2011: (Start)

The matrix of a(n) values

  a(0)

  a(1) a(0)

  a(2) a(1) a(0)

  a(3) a(2) a(1) a(0)

  ....

  a(n) a(n-1) a(n-2) ... a(0)

is the inverse of the matrix

   1

  -1  1

  -1 -1  1

   0 -1 -1  1

  ....

  -d_n  -d_(n-1) -d_(n-2) ... -d_1  1

where d_q = (-1)^(m+1) if q = m(3m-1)/2 = the m-th generalized pentagonal number (A001318), = 0 otherwise. (End)

Equals row sums of triangle A187566. - Gary W. Adamson, Mar 21 2011

Let k > 0 be an integer, and let i_1, i_2, ..., i_k be distinct integers such that 1 <= i_1 < i_2 < ... < i_k. Then, equivalently, a(n) equals the number of partitions of N = n + i_1 + i_2 + ... + i_k in which each i_j (1 <= j <= k) appears as a part at least once. To see this, note that the partitions of N of this class must be in 1-to-1 correspondence with the partitions of n, since N - i_1 - i_2 - ... - i_k = n. - L. Edson Jeffery, Apr 16 2011

a(n) is the number of distinct degree sequences over all free trees having n + 2 nodes.  Take a partition of the integer n, add 1 to each part and append as many 1's as needed so that the total is 2n + 2.  Now we have a degree sequence of a tree with n + 2 nodes.  Example: The partition 3 + 2 + 1 = 6 corresponds to the degree sequence {4, 3, 2, 1, 1, 1, 1, 1} of a tree with 8 vertices. - Geoffrey Critzer, Apr 16 2011

a(n) is number of distinct characteristic polynomials among  n! of permutations matrices size n X n. - Artur Jasinski, Oct 24 2011

Conjecture: starting with offset 1 represents the numbers of ordered compositions of n using the signed (++--++...) terms of A001318 starting (1, 2, -5, -7, 12, 15, ...). - Gary W. Adamson, Apr 04 2013 (this is true by the pentagonal number theorem, Joerg Arndt, Apr 08 2013)

a(n) is also number of terms in expansion of the n-th derivative of log(f(x)). In Mathematica notation: Table[Length[Together[f[x]^n * D[Log[f[x]], {x, n}]]], {n, 1, 20}]. - Vaclav Kotesovec, Jun 21 2013

Conjecture: No a(n) has the form x^m with m > 1 and x > 1. - Zhi-Wei Sun, Dec 02 2013

Partitions of n that contain a part p are the partitions of n - p. Thus, number of partitions of m*n - r that include k*n as a part is A000041(h*n-r), where h = m - k >= 0, n >= 2, 0 <= r < n; see A111295 as an example.  - Clark Kimberling, Mar 03 2014

a(n) is the number of compositions of n into positive parts avoiding the pattern [1, 2]. - Bob Selcoe, Jul 08 2014

Conjecture: For any j there exists k such that all primes p <= A000040(j) are factors of one or more a(n) <= a(k). Growth of this coverage is slow and irregular. k = 1067 covers the first 102 primes, thus slower than A000027. - Richard R. Forberg, Dec 08 2014

a(n) is the number of nilpotent conjugacy classes in the order-preserving, order-decreasing and (order-preserving and order-decreasing) injective transformation semigroups. - Ugbene Ifeanyichukwu, Jun 03 2015

Define a segmented partition a(n,k, <s(1)..s(j)>) to be a partition of n with exactly k parts, with s(j) parts t(j) identical to each other and distinct from all the other parts. Note that n >= k, j <= k, 0 <= s(j) <= k, s(1)t(1) + ... + s(j)t(j) = n and s(1) + ... + s(j) = k. Then there are up to a(k) segmented partitions of n with exactly k parts. - Gregory L. Simay, Nov 08 2015

(End)

From Gregory L. Simay, Nov 09 2015: (Start)

The polynomials for a(n, k, <s(1), ..., s(j)>) have degree j-1.

  a(n, k, <k>) = 1 if n = 0 mod k, = 0 otherwise

  a(rn, rk, <r*s(1), ..., r*s(j)>) = a(n, k, <s(1), ..., s(j)>)

  a(n odd, k, <all s(j) even>) = 0

Established results can be recast in terms of segmented partitions:

  For j(j+1)/2 <= n < (j+1)(j+2)/2, A000009(n) = a(n, 1, <1>) + ... + a(n, j, <j 1's>), j < n

  a(n, k, <j 1's> = a(n - j(j-1)/2, k)

(End)

a(10^20) was computed using the NIST Arb package. It has 11140086260 digits and its head and tail sections are 18381765...88091448. See the Johansson 2015 link. - Stanislav Sykora, Feb 01 2016

Satisfies Benford's law [Anderson-Rolen-Stoehr, 2011]. - N. J. A. Sloane, Feb 08 2017

The partition function p(n) is log-concave for all n>25 [DeSalvo-Pak, 2014]. - Michel Marcus, Apr 30 2019

a(n) is also the dimension of the n-th cohomology of the infinite real Grassmannian with coefficients in Z/2. - Luuk Stehouwer, Jun 06 2021

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PYTHAGORAS, Ramanujan and The Partition Function(Text in Dutch)

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Srinivasa Ramanujan, Congruence Properties Of Partitions

Srinivasa Ramanujan, Congruence Properties Of Partitions

Srinivasa Ramanujan and G. H. Hardy, Une formule asymptotique pour le nombre de partitions de n

J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.

J. D. Rosenhouse, Partitions of Integers

J. D. Rosenhouse, Solutions to Problems

Kate Rudolph, Pattern Popularity in 132-Avoiding Permutations, The Electronic Journal of Combinatorics 20(1)(2013), #P8. [Cited by Shalosh B. Ekhad and Doron Zeilberger, 2014] - N. J. A. Sloane, Mar 31 2014

F. Ruskey, Generate Numerical Partitions

F. Ruskey, The first 284547 partition numbers (52MB compressed file, archived link)

M. Savic, The Partition Function and Ramanujan's 5k+4 Congruence

Zhumagali Shomanov, Combinatorial formula for the partition function, arXiv:1508.03173 [math.CO], 2015.

T. Sillke, Number of integer partitions

R. P. Stanley, A combinatorial miscellany

Yi Wang and Bao-Xuan Zhu, Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences, arXiv preprint arXiv:1303.5595 [math.CO], 2013.

R. L. Weaver, New Congruences for the Partition Function, The Ramanujan Journal 5(1) 2001.

Eric Weisstein's World of Mathematics, Partition, Partition Function P, q-Pochhammer Symbol, and Ramanujan's Identity

West Sussex Grid for Learning, Multicultural Mathematics, Ramanujan's Partition of Numbers

Thomas Wieder, Comment on A000041

Wikipedia, Integer Partition

H. S. Wilf, Lectures on Integer Partitions

Wolfram Research, Generating functions of p(n)

D. J. Wright, Partitions [broken link]

Doron Zeilberger, Noam Zeilberger, Two Questions about the Fractional Counting of Partitions, arXiv:1810.12701 [math.CO], 2018.

Robert M. Ziff, On Cardy's formula for the critical crossing probability in 2d percolation, J. Phys. A. 28, 1249-1255 (1995).

Index entries for "core" sequences

Index entries for related partition-counting sequences

Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Index entries for sequences related to rooted trees

Index entries for sequences related to Benford's law

G.f.: Product_{k>0} 1/(1-x^k) = Sum_{k>= 0} x^k Product_{i = 1..k} 1/(1-x^i) = 1 + Sum_{k>0} x^(k^2)/(Product_{i = 1..k} (1-x^i))^2.

G.f.: 1 + Sum_{n>=1} x^n/(Product_{k>=n} 1-x^k). - Joerg Arndt, Jan 29 2011

a(n) - a(n-1) - a(n-2) + a(n-5) + a(n-7) - a(n-12) - a(n-15) + ... = 0, where the sum is over n-k and k is a generalized pentagonal number (A001318) <= n and the sign of the k-th term is (-1)^([(k+1)/2]). See A001318 for a good way to remember this!

a(n) = (1/n) * Sum_{k=0..n-1} sigma(n-k)*a(k), where sigma(k) is the sum of divisors of k (A000203).

a(n) ~ 1/(4*n*sqrt(3)) * e^(Pi * sqrt(2n/3)) as n -> infinity (Hardy and Ramanujan). See A050811.

a(n) = a(0)*b(n) + a(1)*b(n-2) + a(2)*b(n-4) + ... where b = A000009.

From Jon E. Schoenfield, Aug 17 2014: (Start)

It appears that the above approximation from Hardy and Ramanujan can be refined as

a(n) ~ 1/(4*n*sqrt(3)) * e^(Pi * sqrt(2n/3 + c0 + c1/n^(1/2) + c2/n + c3/n^(3/2) + c4/n^2 + ...)), where the coefficients c0 through c4 are approximately

     c0 = -0.230420145062453320665537

     c1 = -0.0178416569128570889793

     c2 =  0.0051329911273

     c3 = -0.0011129404

     c4 =  0.0009573,

as n -> infinity. (End)

From Vaclav Kotesovec, May 29 2016 (c4 added Nov 07 2016): (Start)

c0 = -0.230420145062453320665536704197233... = -1/36 - 2/Pi^2

c1 = -0.017841656912857088979502135349949... = 1/(6*sqrt(6)*Pi) - sqrt(3/2)/Pi^3

c2 =  0.005132991127342167594576391633559... = 1/(2*Pi^4)

c3 = -0.001112940489559760908236602843497... = 3*sqrt(3/2)/(4*Pi^5) - 5/(16*sqrt(6)*Pi^3)

c4 =  0.000957343284806972958968694349196... = 1/(576*Pi^2) - 1/(24*Pi^4) + 93/(80*Pi^6)

a(n) ~ exp(Pi*sqrt(2*n/3))/(4*sqrt(3)*n) * (1 - (sqrt(3/2)/Pi + Pi/(24*sqrt(6)))/sqrt(n) + (1/16 + Pi^2/6912)/n).

a(n) ~ exp(Pi*sqrt(2*n/3) - (sqrt(3/2)/Pi + Pi/(24*sqrt(6)))/sqrt(n) + (1/24 - 3/(4*Pi^2))/n) / (4*sqrt(3)*n).

(End)

a(n) < exp( (2/3)^(1/2) Pi sqrt(n) ) (Ayoub, p. 197).

G.f.: Product_{m>=1} (1+x^m)^A001511(m). - Vladeta Jovovic, Mar 26 2004

a(n) = Sum_{i=0..n-1} P(i, n-i), where P(x, y) is the number of partitions of x into at most y parts and P(0, y)=1. - Jon Perry, Jun 16 2003

G.f.: Product_{i>=1} Product_{j>=0} (1+x^((2i-1)*2^j))^(j+1). - Jon Perry, Jun 06 2004

G.f. e^(Sum_{k>0} (x^k/(1-x^k)/k)). - Franklin T. Adams-Watters, Feb 08 2006

a(n) = A114099(9*n). - Reinhard Zumkeller, Feb 15 2006

Euler transform of all 1's sequence (A000012). Weighout transform of A001511. - Franklin T. Adams-Watters, Mar 15 2006

a(n) = A027187(n) + A027193(n) = A000701(n) + A046682(n). - Reinhard Zumkeller, Apr 22 2006

Convolved with A152537 gives A000079, powers of 2. - Gary W. Adamson, Dec 06 2008

a(n) = A026820(n, n); a(n) = A108949(n) + A045931(n) + A108950(n) = A130780(n) + A171966(n) - A045931(n) = A045931(n) + A171967(n). - Reinhard Zumkeller, Jan 21 2010

a(n) = Tr(n)/(24*n-1) = A183011(n)/A183010(n), n>=1. See the Bruinier-Ono paper in the Links. - Omar E. Pol, Jan 23 2011

From Jerome Malenfant, Feb 14 2011: (Start)

a(n) = determinant of the n X n Toeplitz matrix:

   1  -1

   1   1  -1

   0   1   1  -1

   0   0   1   1  -1

  -1   0   0   1   1  -1

   . . .

  d_n  d_(n-1) d_(n-2)...1

where d_q = (-1)^(m+1) if q = m(3m-1)/2 = p_m, the m-th generalized pentagonal number (A001318), otherwise d_q = 0. Note that the 1's run along the diagonal and the -1's are on the superdiagonal. The (n-1) row (not written) would end with ... 1 -1. (End)

Empirical: let F*(x) = Sum_{n=0..infinity} p(n)*exp(-Pi*x*(n+1)), then F*(2/5) = 1/sqrt(5) to a precision of 13 digits.

F*(4/5) = 1/2+3/2/sqrt(5)-sqrt(1/2*(1+3/sqrt(5))) to a precision of 28 digits. These are the only values found for a/b when a/b is from F60, Farey fractions up to 60. The number for F*(4/5) is one of the real roots of 25*x^4 - 50*x^3 - 10*x^2 - 10*x + 1. Note here the exponent (n+1) compared to the standard notation with n starting at 0. - Simon Plouffe, Feb 23 2011

The constant (2^(7/8)*GAMMA(3/4))/(exp(Pi/6)*Pi^(1/4)) = 1.0000034873... when expanded in base exp(4*Pi) will give the first 52 terms of a(n), n>0, the precision needed is 300 decimal digits. - Simon Plouffe, Mar 02 2011

a(n) = A035363(2n). - Omar E. Pol, Nov 20 2009

G.f.: A(x)=1+x/(G(0)-x); G(k) = 1 + x - x^(k+1) - x*(1-x^(k+1))/G(k+1); (continued fraction Euler's kind, 1-step ). - Sergei N. Gladkovskii, Jan 25 2012

Convolution of A010815 with A000712. - Gary W. Adamson, Jul 20 2012

G.f.: 1 + x*(1 - G(0))/(1-x) where G(k) = 1 - 1/(1-x^(k+1))/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 22 2013

G.f.: Q(0) where Q(k) = 1 + x^(4*k+1)/( (x^(2*k+1)-1)^2 - x^(4*k+3)*(x^(2*k+1)-1)^2/( x^(4*k+3) + (x^(2*k+2)-1)^2/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Feb 16 2013

a(n) = 24*spt(n) + 12*N_2(n) - Tr(n) = 24*A092269(n) + 12*A220908(n) - A183011(n), n >= 1. - Omar E. Pol, Feb 17 2013

G.f.: 1/(x; x)_{inf} where (a; q)_k is the q-Pochhammer symbol. - Vladimir Reshetnikov, Apr 24 2013

a(n) = A066186(n)/n, n >= 1. - Omar E. Pol, Aug 16 2013

From Peter Bala, Dec 23 2013: (Start)

a(n-1) = Sum_{parts k in all partitions of n} mu(k), where mu(k) is the arithmetical Möbius function (see A008683).

Let P(2,n) denote the set of partitions of n into parts k >= 2. Then a(n-2) = -Sum_{parts k in all partitions in P(2,n)} mu(k).

n*( a(n) - a(n-1) ) = Sum_{parts k in all partitions in P(2,n)} k (see A138880).

Let P(3,n) denote the set of partitions of n into parts k >= 3. Then

a(n-3) = (1/2)*Sum_{parts k in all partitions in P(3,n)} phi(k), where phi(k) is the Euler totient function (see A000010). Using this result and Mertens's theorem on the average order of the phi function, we can find an approximate 3-term recurrence for the partition function: a(n) ~ a(n-1) + a(n-2) + (Pi^2/(3*n) - 1)*a(n-3). For example, substituting the values a(47) = 124754, a(48) = 147273 and a(49) = 173525 into the recurrence gives the approximation a(50) ~ 204252.48... compared with the true value a(50) = 204226. (End)

a(n) = Sum_{k=1..n+1} (-1)^(n+1-k)*A000203(k)*A002040(n+1-k). - Mircea Merca, Feb 27 2014

a(n) = A240690(n) + A240690(n+1), n >= 1. - Omar E. Pol, Mar 16 2015

From Gary W. Adamson, Jun 22 2015: (Start)

A production matrix for the sequence with offset 1 is M, an infinite n x n matrix of the following form:

  a, 1, 0, 0, 0, 0, ...

  b, 0, 1, 0, 0, 0, ...

  c, 0, 0, 1, 0, 0, ...

  d, 0, 0, 0, 1, 0, ...

.

.

... such that (a, b, c, d, ...) is the signed version of A080995 with offset 1: (1,1,0,0,-1,0,-1,...)

and a(n) is the upper left term of M^n.

This operation is equivalent to the g.f. (1 + x + 2x^2 + 3x^3 + 5x^4 + ...) = 1/(1 - x - x^2 + x^5 + x^7 - x^12 - x^15 + x^22 + ...). (End)

G.f.: x^(1/24)/eta(log(x)/(2 Pi i)). - Thomas Baruchel, Jan 09 2016, after Michael Somos (after Richard Dedekind).

a(n) = Sum_{k=-inf..+inf} (-1)^k a(n-k(3k-1)/2) with a(0)=1 and a(negative)=0. The sum can be restricted to the (finite) range from k = (1-sqrt(1-24n))/6 to (1+sqrt(1-24n))/6), since all terms outside this range are zero. - Jos Koot, Jun 01 2016

G.f.: (conjecture) (r(x) * r(x^2) * r(x^4) * r(x^8) * ...) where r(x) is A000009: (1, 1, 1, 2, 2, 3, 4, ...). - Gary W. Adamson, Sep 18 2016; Doron Zeilberger observed today that "This follows immediately from Euler's formula 1/(1-z) = (1+z)*(1+z^2)*(1+z^4)*(1+z^8)*..." Gary W. Adamson, Sep 20 2016

a(n) ~ 2*Pi * BesselI(3/2, sqrt(24*n-1)*Pi/6) / (24*n-1)^(3/4). - Vaclav Kotesovec, Jan 11 2017

G.f.: Product_{k>=1} (1 + x^k)/(1 - x^(2*k)). - Ilya Gutkovskiy, Jan 23 2018

a(n) = p(1, n) where p(k, n) = p(k+1, n) + p(k, n-k) if k < n, 1 if k = n, and 0 if k > n. p(k, n) is the number of partitions of n into parts >= k. - Lorraine Lee, Jan 28 2020

Sum_{n>=1} 1/a(n) = A078506. - Amiram Eldar, Nov 01 2020

Sum_{n>=0} a(n)/2^n = A065446. - Amiram Eldar, Jan 19 2021

From Simon Plouffe, Mar 12 2021: (Start)

Empirical: Sum_{n>=0}, a(n)/exp(Pi*(n-1)) = 2^(3/8)*Gamma(3/4)/(Pi^(1/4)*exp(Pi/24)).

Empirical: Sum_{n>=0}, a(n)/exp(2*Pi*(n-1)) = 2^(1/2)*Gamma(3/4)/(Pi^(1/4)*exp(Pi/12)). (End) [These are the reciprocals of phi(exp(-Pi)) (A259148) and phi(exp(-2*Pi)) (A259149), where phi(q) is the Euler modular function. See B. C. Berndt (RLN, Vol. V, p. 326), and formulas (13) and (14) in I. Mező, 2013. - Peter Luschny, Mar 13 2021]

a(n) = A000009(n)+A035363(n)+A006477(n). - R. J. Mathar, Feb 01 2022

a(5) = 7 because there are seven partitions of 5, namely: {1, 1, 1, 1, 1}, {2, 1, 1, 1}, {2, 2, 1}, {3, 1, 1}, {3, 2}, {4, 1}, {5}. - Bob Selcoe, Jul 08 2014

G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + ...

G.f. = 1/q + q^23 + 2*q^47 + 3*q^71 + 5*q^95 + 7*q^119 + 11*q^143 + 15*q^167 + ...

From Gregory L. Simay, Nov 08 2015: (Start)

There are up to a(4)=5 segmented partitions of the partitions of n with exactly 4 parts. They are a(n,4, <4>), a(n,4,<3,1>), a(n,4,<2,2>), a(n,4,<2,1,1>), a(n,4,<1,1,1,1>).

The partition 8,8,8,8 is counted in a(32,4,<4>).

The partition 9,9,9,5 is counted in a(32,4,<3,1>).

The partition 11,11,5,5 is counted in a(32,4,<2,2>).

The partition 13,13,5,1 is counted in a(32,4,<2,1,1>).

The partition 14,9,6,3 is counted in a(32,4,<1,1,1,1>).

a(n odd,4,<2,2>) = 0.

a(12, 6, <2,2,2>) = a(6,3,<1,1,1>) = a(6-3,3) = a(3,3) = 1. The lone partition is 3,3,2,2,1,1.

(End)

A000041 := n -> combinat:-numbpart(n): [seq(A000041(n), n=0..50)]; # Warning: Maple 10 and 11 give incorrect answers in some cases: A110375.

spec := [B, {B=Set(Set(Z, card>=1))}, unlabeled ];

[seq(combstruct[count](spec, size=n), n=0..50)];

with(combstruct):ZL0:=[S, {S=Set(Cycle(Z, card>0))}, unlabeled]: seq(count(ZL0, size=n), n=0..45); # Zerinvary Lajos, Sep 24 2007

G:={P=Set(Set(Atom, card>0))}: combstruct[gfsolve](G, labeled, x); seq(combstruct[count]([P, G, unlabeled], size=i), i=0..45); # Zerinvary Lajos, Dec 16 2007

# Using the function EULER from Transforms (see link at the bottom of the page).

1, op(EULER([seq(1, n=1..49)])); # Peter Luschny, Aug 19 2020

Table[ PartitionsP[n], {n, 0, 45}]

a[ n_] := SeriesCoefficient[ q^(1/24) / DedekindEta[ Log[q] / (2 Pi I)], {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)

a[ n_] := SeriesCoefficient[ 1 / Product[ 1 - x^k, {k, n}], {x, 0, n}]; (* Michael Somos, Jul 11 2011 *)

CoefficientList[1/QPochhammer[q] + O[q]^100, q] (* Jean-François Alcover, Nov 25 2015 *)

(Magma) a:= func< n | NumberOfPartitions(n) >; [ a(n) : n in [0..10]];

(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / eta(x + x * O(x^n)), n))};

(PARI) /* The Hardy-Ramanujan-Rademacher exact formula in PARI is as follows (this is no longer necessary since it is now built in to the numbpart command): */

Psi(n, q) = local(a, b, c); a=sqrt(2/3)*Pi/q; b=n-1/24; c=sqrt(b); (sqrt(q)/(2*sqrt(2)*b*Pi))*(a*cosh(a*c)-(sinh(a*c)/c))

L(n, q) = if(q==1, 1, sum(h=1, q-1, if(gcd(h, q)>1, 0, cos((g(h, q)-2*h*n)*Pi/q))))

g(h, q) = if(q<3, 0, sum(k=1, q-1, k*(frac(h*k/q)-1/2)))

part(n) = round(sum(q=1, max(5, 0.5*sqrt(n)), L(n, q)*Psi(n, q)))

/* Ralf Stephan, Nov 30 2002, fixed by Vaclav Kotesovec, Apr 09 2018 */

(PARI) {a(n) = numbpart(n)};

(PARI) {a(n) = if( n<0, 0, polcoeff( sum( k=1, sqrtint(n), x^k^2 / prod( i=1, k, 1 - x^i, 1 + x * O(x^n))^2, 1), n))};

(PARI) f(n)= my(v, i, k, s, t); v=vector(n, k, 0); v[n]=2; t=0; while(v[1]<n, i=2; while(v[i]==0, i++); v[i]--; s=sum(k=i, n, k*v[k]); while(i>1, i--; s+=i*(v[i]=(n-s)\i)); t++); t \\ Thomas Baruchel, Nov 07 2005

(PARI) a(n)=if(n<0, 0, polcoeff(exp(sum(k=1, n, x^k/(1-x^k)/k, x*O(x^n))), n)) \\ Joerg Arndt, Apr 16 2010

(MuPAD) combinat::partitions::count(i) $i=0..54 // Zerinvary Lajos, Apr 16 2007

(Sage) [number_of_partitions(n) for n in range(46)]  # Zerinvary Lajos, May 24 2009

(Sage)

@CachedFunction

def A000041(n):

    if n == 0: return 1

    S = 0; J = n-1; k = 2

    while 0 <= J:

        T = A000041(J)

        S = S+T if is_odd(k//2) else S-T

        J -= k if is_odd(k) else k//2

        k += 1

    return S

[A000041(n) for n in range(50)]  # Peter Luschny, Oct 13 2012

(Sage) # uses[EulerTransform from A166861]

a = BinaryRecurrenceSequence(1, 0)

b = EulerTransform(a)

print([b(n) for n in range(50)]) # Peter Luschny, Nov 11 2020

(Haskell)

import Data.MemoCombinators (memo2, integral)

a000041 n = a000041_list !! n

a000041_list = map (p' 1) [0..] where

   p' = memo2 integral integral p

   p _ 0 = 1

   p k m = if m < k then 0 else p' k (m - k) + p' (k + 1) m

-- Reinhard Zumkeller, Nov 03 2015, Nov 04 2013

(Maxima) num_partitions(60, list); /* Emanuele Munarini, Feb 24 2014 */

(GAP) List([1..10], n->Size(OrbitsDomain(SymmetricGroup(IsPermGroup, n), SymmetricGroup(IsPermGroup, n), \^))); # Attila Egri-Nagy, Aug 15 2014

(Perl) use ntheory ":all"; my @p = map { partitions($_) } 0..100; say "[@p]"; # Dana Jacobsen, Sep 06 2015

(Racket)

#lang racket

; SUM(k, -inf, +inf) (-1)^k p(n-k(3k-1)/2)

; For k outside the range (1-(sqrt(1-24n))/6 to (1+sqrt(1-24n))/6) argument n-k(3k-1)/2 < 0.

; Therefore the loops below are finite. The hash avoids repeated identical computations.

(define (p n) ; Nr of partitions of n.

(hash-ref h n

  (λ ()

   (define r

    (+

     (let loop ((k 1) (n (sub1 n)) (s 0))

      (if (< n 0) s

       (loop (add1 k) (- n (* 3 k) 1) (if (odd? k) (+ s (p n)) (- s (p n))))))

     (let loop ((k -1) (n (- n 2)) (s 0))

      (if (< n 0) s

       (loop (sub1 k) (+ n (* 3 k) -2) (if (odd? k) (+ s (p n)) (- s (p n))))))))

   (hash-set! h n r)

   r)))

(define h (make-hash '((0 . 1))))

; (for ((k (in-range 0 50))) (printf "~s, " (p k))) runs in a moment.

; Jos Koot, Jun 01 2016

(Python)

from sympy.ntheory import npartitions

print([npartitions(i) for i in range(101)])  # Indranil Ghosh, Mar 17 2017

(Julia) # DedekindEta is defined in A000594

A000041List(len) = DedekindEta(len, -1)

A000041List(50) |> println # Peter Luschny, Mar 09 2018

Cf. A000009, A000079, A000203, A001318, A008284, A065446, A078506, A113685, A132311, A145006, A145007, A147843, A152537, A168532, A173238, A173239, A173241, A173304, A174065, A174066, A174068, A176202.

For successive differences see A002865, A053445, A072380, A081094, A081095.

Antidiagonal sums of triangle A092905. a(n) = A054225(n,0).

Boustrophedon transforms: A000733, A000751.

Cf. A167376 (complement), A061260 (multisets).

Sequence in context: A092885 A330643 A213598 * A280662 A218027 A241729

Adjacent sequences:  A000038 A000039 A000040 * A000042 A000043 A000044

core,easy,nonn,nice

N. J. A. Sloane

Additional comments from Ola Veshta (olaveshta(AT)my-deja.com), Feb 28 2001

Additional comments from Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001

approved