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A341265 Coefficient of x^(2*n) in (-1 + Product_{k>=1} 1 / (1 + x^k))^n.

%I A341265 #13 Feb 20 2021 06:20:14
%S A341265 1,0,2,3,10,25,71,203,562,1650,4667,13673,39427,115440,336639,987628,
%T A341265 2898658,8529257,25134200,74173606,219207815,648546314,1921045953,
%U A341265 5695642513,16902924883,50203798050,149229323544,443895849894,1321292939459,3935377071154,11728037768186
%N A341265 Coefficient of x^(2*n) in (-1 + Product_{k>=1} 1 / (1 + x^k))^n.
%H A341265 Alois P. Heinz, <a href="/A341265/b341265.txt">Table of n, a(n) for n = 0..1000</a>
%F A341265 a(n) = A341279(2n,n).
%F A341265 a(n) ~ c * d^n / sqrt(n), where d = 3.03044218957412050685579849718626198523346... and c = 0.2319377657497495246637662111041144... - _Vaclav Kotesovec_, Feb 20 2021
%p A341265 g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
%p A341265       [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
%p A341265     end:
%p A341265 b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, g(n+1),
%p A341265       (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A341265     end:
%p A341265 a:= n-> b(n$2):
%p A341265 seq(a(n), n=0..30);  # _Alois P. Heinz_, Feb 07 2021
%t A341265 Table[SeriesCoefficient[(-1 + 1/QPochhammer[-x, x])^n, {x, 0, 2 n}], {n, 0, 30}]
%t A341265 A[n_, k_] := A[n, k] = If[n == 0, 1, -k Sum[A[n - j, k] Sum[Mod[d, 2] d, {d, Divisors[j]}], {j, 1, n}]/n]; T[n_, k_] := Sum[(-1)^i Binomial[k, i] A[n, k - i], {i, 0, k}]; Table[T[2 n, n], {n, 0, 30}]
%Y A341265 Cf. A000700, A081362, A255526, A324595, A338463, A340987, A341241, A341243, A341244, A341245, A341246, A341247, A341251, A341253, A341263, A341279.
%K A341265 nonn
%O A341265 0,3
%A A341265 _Ilya Gutkovskiy_, Feb 07 2021