A341263 Coefficient of x^(2*n) in (-1 + Product_{k>=1} (1 - x^k))^n.

1, -1, 1, -1, -3, 19, -65, 181, -419, 755, -749, -1530, 12255, -47477, 141065, -343526, 660941, -770917, -911369, 9721976, -40135713, 124134772, -313463842, 631382751, -824406065, -492101356, 8192253811, -35948431288, 115087580857, -299576625051, 627027769120, -894734468883

Offset: 0

Author

Ilya Gutkovskiy, Feb 07 2021

Keywords

• sign

Crossrefs

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

Programs

Maple

g:= proc(n) option remember; `if`(n=0, 1, add(add(
     -d, d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, g(n+1),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..31);  # Alois P. Heinz, Feb 07 2021

Mathematica

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