A341263 - OEIS

%I #9 Feb 07 2021 18:38:36

%S 1,-1,1,-1,-3,19,-65,181,-419,755,-749,-1530,12255,-47477,141065,

%T -343526,660941,-770917,-911369,9721976,-40135713,124134772,

%U -313463842,631382751,-824406065,-492101356,8192253811,-35948431288,115087580857,-299576625051,627027769120,-894734468883

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

%p g:= proc(n) option remember; `if`(n=0, 1, add(add(

%p      -d, d=numtheory[divisors](j))*g(n-j), j=1..n)/n)

%p     end:

%p b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, g(n+1),

%p       (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..31);  # _Alois P. Heinz_, Feb 07 2021

%t Table[SeriesCoefficient[(-1 + QPochhammer[x, x])^n, {x, 0, 2 n}], {n, 0, 31}]

%t A[n_, k_] := A[n, k] = If[n == 0, 1, -k Sum[A[n - j, k] DivisorSigma[1, j], {j, 1, n}]/n]; T[n_, k_] := Sum[(-1)^i Binomial[k, i] A[n, k - i], {i, 0, k}];

%t Table[T[2 n, n], {n, 0, 31}]

%Y Cf. A001482, A001483, A001484, A001485, A001486, A001487, A001488, A008705, A010815, A047654, A047655, A324595, A340987, A341265.

%K sign

%O 0,5

%A _Ilya Gutkovskiy_, Feb 07 2021