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%I #19 Jan 23 2023 13:14:26
%S 0,0,0,1,2,4,7,11,16,23,32,43,57,74,95,121,152,189,234,287,350,425,
%T 513,616,737,878,1042,1233,1454,1709,2004,2343,2732,3179,3690,4274,
%U 4941,5700,6563,7544,8656,9915,11340,12949,14764,16811,19114,21703,24612,27875,31532,35628,40209
%N Expansion of -1/(1 - x)^2 + (1/(1 - x))*Product_{k>=1} (1 + x^k).
%C Partial sums of A111133.
%H Michael De Vlieger, <a href="/A317910/b317910.txt">Table of n, a(n) for n = 0..10000</a>
%H Riccardo Aragona, Roberto Civino, and Norberto Gavioli, <a href="https://arxiv.org/abs/2301.06347">A modular idealizer chain and unrefinability of partitions with repeated parts</a>, arXiv:2301.06347 [math.RA], 2023.
%H Riccardo Aragona, Roberto Civino, Norberto Gavioli, Carlo Maria Scoppola, <a href="https://arxiv.org/abs/2008.13423">A Chain of Normalizers in the Sylow _2-subgroups of the symmetric group on 2^n letters</a>, arXiv:2008.13423 [math.GR], 2020.
%H Riccardo Aragona, Roberto Civino, Norberto Gavioli, and Carlo Maria Scoppola, <a href="https://arxiv.org/abs/2009.11149">Rigid commutators and a normalizer chain</a>, arXiv:2009.11149 [math.GR], 2020.
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F G.f.: -1/(1 - x)^2 + (1/(1 - x))*Product_{k>=1} 1/(1 - x^(2*k-1)).
%F a(n) = A036469(n) - n - 1.
%F a(n) ~ 3^(1/4) * exp(Pi*sqrt(n/3)) / (2*Pi*n^(1/4)). - _Vaclav Kotesovec_, Aug 21 2018
%p a:=series(-1/(1-x)^2+(1/(1-x))*mul((1 + x^k),k=1..100),x=0,53): seq(coeff(a,x,n),n=0..52); # _Paolo P. Lava_, Apr 02 2019
%t nmax = 52; CoefficientList[Series[-1/(1 - x)^2 + 1/(1 - x) Product[1 + x^k, {k, 1, nmax}], {x, 0, nmax}], x] (* or *)
%t nmax = 52; CoefficientList[Series[1/((1 - x) QPochhammer[x, x^2]) - 1/(1 - x)^2, {x, 0, nmax}], x] (* or *)
%t Table[Sum[PartitionsQ[k] - 1, {k, 0, n}] , {n, 0, 52}]
%Y Cf. A026906, A036469, A058682, A111133.
%K nonn
%O 0,5
%A _Ilya Gutkovskiy_, Aug 10 2018
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