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

1, 0, 2, 3, 10, 25, 71, 203, 562, 1650, 4667, 13673, 39427, 115440, 336639, 987628, 2898658, 8529257, 25134200, 74173606, 219207815, 648546314, 1921045953, 5695642513, 16902924883, 50203798050, 149229323544, 443895849894, 1321292939459, 3935377071154, 11728037768186

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

a(n) = A341279(2n,n).

a(n) ~ c * d^n / sqrt(n), where d = 3.03044218957412050685579849718626198523346... and c = 0.2319377657497495246637662111041144... - Vaclav Kotesovec, Feb 20 2021

Maple

g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
      [1+irem(d, 4)], 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..30);  # Alois P. Heinz, Feb 07 2021

Mathematica

Table[SeriesCoefficient[(-1 + 1/QPochhammer[-x, x])^n, {x, 0, 2 n}], {n, 0, 30}]
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}]

Crossrefs

Cf. A000700, A081362, A255526, A324595, A338463, A340987, A341241, A341243, A341244, A341245, A341246, A341247, A341251, A341253, A341263, A341279.

Sequence in context: A123029 A103018 A246437 * A378424 A005158 A370608

Adjacent sequences: A341262 A341263 A341264 * A341266 A341267 A341268

Keyword

nonn

Author

Ilya Gutkovskiy, Feb 07 2021

Status

approved