0,5

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

For n > 0, also the number of partitions of n whose greatest part is even. [Edited by Gus Wiseman, Jan 05 2021]

Number of partitions of n+1 into an odd number of parts, the least being 1.

Also the number of partitions of n such that the number of even parts has the same parity as the number of odd parts; see Comments at A027193. - Clark Kimberling, Feb 01 2014, corrected Jan 06 2021

Suppose that c(0) = 1, that c(1), c(2), ... are indeterminates, that d(0) = 1, and that d(n) = -c(n) - c(n-1)*d(1) - ... - c(0)*d(n-1).  When d(n) is expanded as a polynomial in c(1), c(2),..,c(n), the terms are of the form H*c(i_1)*c(i_2)*...*c(i_k).  Let P(n) = [c(i_1), c(i_2), ..., c(i_k)], a partition of n.  Then H is negative if P has an odd number of parts, and H is positive if P has an even number of parts.  That is, d(n) has A027193(n) negative coefficients, A027187(n) positive coefficients, and A000041 terms.  The maximal coefficient in d(n), in absolute value, is A102462(n). - Clark Kimberling, Dec 15 2016

Roland Bacher, P De La Harpe, Conjugacy growth series of some infinitely generated groups. 2016. hal-01285685v2; https://hal.archives-ouvertes.fr/hal-01285685/document

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; See p. 8, (7.323) and p. 39, Example 7.

T. D. Noe, Table of n, a(n) for n = 0..999

George E. Andrews, David Newman, The Minimal Excludant in Integer Partitions, J. Int. Seq., Vol. 23 (2020), Article 20.2.3.

N. J. Fine, Problem 4314, Amer. Math. Monthly, Vol. 57, 1950, 421-423.

Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function p_e(n).

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

a(n) = (A000041(n) + (-1)^n * A000700(n))/2.

a(n) = p(n) - p(n-1) + p(n-4) - p(n-9) + ... where p(n) is the number of unrestricted partitions of n, A000041. [Fine] - David Callan, Mar 14 2004

From Bill Gosper, Jun 25 2005: (Start)

G.f.: A(q) = Sum_{n >= 0} a(n) q^n = 1 + q^2 + q^3 + 3 q^4 + 3 q^5 + 6 q^6 + ...

= Sum_{n >= 0} q^(2n)/(q; q)_{2n}

= ((Product_{k >= 1} 1/(1-q^k) + (Product_{k >= 1} 1/(1+q^k))/2.

Also, let B(q) = Sum_{n >= 0} A027193(n) q^n = q + q^2 + 2 q^3 + 2 q^4 + 4 q^5 + 5 q^6 + ...

Then B(q) = Sum_{n >= 0} q^(2n+1)/(q; q)_{2n+1} = ((Product_{k >= 1} 1/(1-q^k) - (Product_{k >= 1} 1/(1+q^k))/2.

Also we have the following identity involving 2 X 2 matrices:

Product_{k >= 1} [ 1/(1-q^2k) q^k/(1-q^2k / q^k/(1-q^2k) 1/(1-q^2k) ]

= [ A(q) B(q) / B(q) A(q) ]. (End)

a(2*n) = A046682(2*n), a(2*n+1) = A000701(2*n+1); a(n) = A000041(n)-A027193(n). - Reinhard Zumkeller, Apr 22 2006

Expansion of (1 + phi(-q)) / (2 * f(-q)) where phi(), f() are Ramanujan theta functions. - Michael Somos, Aug 19 2006

G.f.: (Sum_{k>=0} (-1)^k * x^(k^2)) / (Product_{k>0} (1 - x^k)). - Michael Somos, Aug 19 2006

a(n) = A338914(n) + A096373(n). - Gus Wiseman, Jan 06 2021

G.f. = 1 + x^2 + x^3 + 3*x^4 + 3*x^5 + 6*x^6 + 7*x^7 + 12*x^8 + 14*x^9 + 22*x^10 + ...

From Gus Wiseman, Jan 05 2021: (Start)

The a(2) = 1 through a(8) = 12 partitions into an even number of parts are the following. The Heinz numbers of these partitions are given by A028260.

  (11)  (21)  (22)    (32)    (33)      (43)      (44)

              (31)    (41)    (42)      (52)      (53)

              (1111)  (2111)  (51)      (61)      (62)

                              (2211)    (2221)    (71)

                              (3111)    (3211)    (2222)

                              (111111)  (4111)    (3221)

                                        (211111)  (3311)

                                                  (4211)

                                                  (5111)

                                                  (221111)

                                                  (311111)

                                                  (11111111)

The a(2) = 1 through a(8) = 12 partitions whose greatest part is even are the following. The Heinz numbers of these partitions are given by A244990.

  (2)  (21)  (4)    (41)    (6)      (43)      (8)

             (22)   (221)   (42)     (61)      (44)

             (211)  (2111)  (222)    (421)     (62)

                            (411)    (2221)    (422)

                            (2211)   (4111)    (431)

                            (21111)  (22111)   (611)

                                     (211111)  (2222)

                                               (4211)

                                               (22211)

                                               (41111)

                                               (221111)

                                               (2111111)

(End)

f[n_] := Length[Select[IntegerPartitions[n], IntegerQ[First[#]/2] &]]; Table[f[n], {n, 1, 30}] (* Clark Kimberling, Mar 13 2012 *)

a[ n_] := SeriesCoefficient[ (1 + EllipticTheta[ 4, 0, x]) / (2 QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, May 06 2015 *)

a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[n], EvenQ[Length @ #] &]]; (* Michael Somos, May 06 2015 *)

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum( k=0, sqrtint(n), (-x)^k^2, A) / eta(x + A), n))}; /* Michael Somos, Aug 19 2006 */

(PARI) q='q+O('q^66); Vec( (1/eta(q)+eta(q)/eta(q^2))/2 ) \\ Joerg Arndt, Mar 23 2014

Cf. A000701, A046682, A026838, A102462.

The Heinz numbers of these partitions are A028260.

The odd version is A027193.

The strict case is A067661.

The case of even sum as well as length is A236913 (the even bisection).

Other cases of even length:

- A024430 counts set partitions of even length.

- A034008 counts compositions of even length.

- A052841 counts ordered set partitions of even length.

- A174725 counts ordered factorizations of even length.

- A332305 counts strict compositions of even length

- A339846 counts factorizations of even length.

A000009 counts partitions into odd parts, ranked by A066208.

A026805 counts partitions whose least part is even.

A072233 counts partitions by sum and length.

A101708 counts partitions of even positive rank.

Cf. A000700, A026424, A058696, A096373, A244990, A300061.

Sequence in context: A088571 A325834 A241832 * A056508 A050065 A298732

Adjacent sequences:  A027184 A027185 A027186 * A027188 A027189 A027190

nonn

Clark Kimberling

Offset changed to 0 by Michael Somos, Jul 24 2012

approved