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%I #46 Apr 25 2022 08:02:13
%S 1,1,1,0,0,1,2,1,0,0,2,2,1,0,2,3,2,0,2,4,4,0,1,4,6,2,1,4,8,4,2,4,10,6,
%T 2,3,12,10,4,2,13,14,8,2,14,18,12,2,14,22,18,3,14,26,26,6,14,29,34,10,
%U 14,32,44,16,14,34,56,26,16,34,67,38,20,34,78,52,26
%N Coefficient of x^n in 1/((1-x)(1+x^3)(1-x^5)(1+x^7)(1-x^9)...).
%C By Theorem 1 of Craig, the values a(n) in this list are known to be nonnegative. Combined with Theorem 2 of Seo and Yee, this shows that a(n) = |number of partitions of n into odd parts with an odd index minus the number of partitions of n into odd parts with an even index|. - _William Craig_, Dec 31 2021
%D Seunghyun Seo and Ae Ja Yee, Index of seaweed algebras and integer partitions, Electronic Journal of Combinatorics, 27:1 (2020), #P1.47. See Conjecture 1 and Theorem 2.
%H Vaclav Kotesovec, <a href="/A300574/b300574.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..2000 from Gus Wiseman)
%H Vincent E. Coll, Andrew W. Mayers, and Nick W. Mayers, <a href="https://arxiv.org/abs/1809.09271">Statistics on integer partitions arising from seaweed algebras</a>, arXiv:1809.09271 [math.CO], 2018.
%H William Craig, <a href="https://arxiv.org/abs/2112.09269">Seaweed Algebras and the Index Statistic for Partitions</a>, arXiv:2112.09269 [math.CO], 2021.
%H Vaclav Kotesovec, <a href="/A300574/a300574.jpg">Graph - the asymptotic ratio (10000 terms)</a>
%H Vaclav Kotesovec, <a href="/A300574/a300574_1.jpg">Graph - the asymptotic ratio (100000 terms)</a>
%F O.g.f.: Product_{n >= 0} 1/(1 - (-1)^n x^(2n+1)).
%F a(n) = Sum (-1)^k where the sum is over all integer partitions of n into odd parts and k is the number of parts not congruent to 1 modulo 4.
%F a(n) has average order Gamma(1/4) * exp(sqrt(n/3)*Pi/2) / (2^(9/4) * 3^(1/8) * Pi^(3/4) * n^(5/8)). - _Vaclav Kotesovec_, Jun 04 2019
%t CoefficientList[Series[1/QPochhammer[x, -x^2], {x, 0, 100}], x]
%t nmax = 100; CoefficientList[Series[Product[1/((1+x^(4*k-1))*(1-x^(4*k-3))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 04 2019 *)
%Y Cf. A000009, A010815, A027193, A067659, A078408, A081362, A099323, A220418, A290261, A292043, A292137, A298118, A300575.
%K nonn,look,nice
%O 0,7
%A _Gus Wiseman_, Mar 08 2018
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