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A000716
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Number of partitions of n into parts of 3 kinds.
(Formerly M2788 N1123)
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20
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1, 3, 9, 22, 51, 108, 221, 429, 810, 1479, 2640, 4599, 7868, 13209, 21843, 35581, 57222, 90882, 142769, 221910, 341649, 521196, 788460, 1183221, 1762462, 2606604, 3829437, 5590110, 8111346, 11701998, 16790136, 23964594, 34034391, 48104069, 67679109, 94800537, 132230021, 183686994, 254170332
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OFFSET
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0,2
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COMMENTS
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A000712: (1, 2, 5, 10, 20, 36, ...) = A000716 convolved with A010815. - Gary W. Adamson, Oct 26 2008
It appears that the partial sums give A210843. - Omar E. Pol, Jun 18 2012
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REFERENCES
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H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 122.
Moreno, Carlos J., Partitions, congruences and Kac-Moody Lie algebras. Preprint, 37pp., no date. See Table I.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 501 terms from T. D. Noe)
Roland Bacher, P. De La Harpe, Conjugacy growth series of some infinitely generated groups, 2016, hal-01285685v2.
Victor J. W. Guo and Jiang Zeng, Two truncated identities of Gauss, arXiv 1205.4340 [math.CO], 2012. - N. J. A. Sloane, Oct 10 2012
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 391
Vladimir P. Kostov, Asymptotic expansions of zeros of a partial theta function, arXiv:1504.00883 [math.CA], 2015.
V. P. Kostov, Stabilization of the asymptotic expansions of the zeros of a partial theta function, arXiv preprint arXiv:1510.02584 [math.CA], 2015.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 8.
P. Nataf, M. Lajkó, A. Wietek, K. Penc, F. Mila, A. M. Läuchli, Chiral spin liquids in triangular lattice SU (N) fermionic Mott insulators with artificial gauge fields, arXiv preprint arXiv:1601.00958 [cond-mat.quant-gas], 2016.
N. J. A. Sloane, Transforms
Index entries for expansions of Product_{k >= 1} (1-x^k)^m
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FORMULA
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G.f.: Product_{m>=1} 1/(1-x^m)^3.
EULER transform of 3, 3, 3, 3, 3, 3, 3, 3, ...
a(0)=1, a(n) = 1/n*Sum_{k=0..n-1} 3*a(k)*sigma_1(n-k). - Joerg Arndt, Feb 05 2011
a(n) ~ exp(Pi * sqrt(2*n)) / (8 * sqrt(2) * n^(3/2)) * (1 - (3/Pi + Pi/8) / sqrt(2*n)). - Vaclav Kotesovec, Feb 28 2015, extended Jan 16 2017
G.f.: exp(3*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018
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MAPLE
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with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(add(d*3, d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..40); # Alois P. Heinz, May 20 2013
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MATHEMATICA
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a[0] = 1; a[n_] := a[n] = 1/n*Sum[3*a[k]*DivisorSigma[1, n-k], {k, 0, n-1}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 03 2014, after Joerg Arndt *)
(1/QPochhammer[q]^3 + O[q]^40)[[3]] (* Vladimir Reshetnikov, Nov 21 2016 *)
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PROG
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(PARI) Vec(1/eta('x+O('x^66))^3) \\ Joerg Arndt, Apr 28 2013
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CROSSREFS
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Cf. A000712, A000713, A010815.
Column 3 of A144064.
Sequence in context: A160526 A121589 A227454 * A001628 A099166 A222083
Adjacent sequences: A000713 A000714 A000715 * A000717 A000718 A000719
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Extended with formula from Christian G. Bower, Apr 15 1998
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STATUS
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approved
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