A304966 Expansion of Product_{k>=1} 1/(1 - x^k)^(p(k)-1), where p(k) = number of partitions of k (A000041).

1, 0, 1, 2, 5, 8, 18, 30, 61, 107, 203, 358, 663, 1162, 2093, 3666, 6481, 11258, 19652, 33874, 58464, 100046, 171032, 290563, 492745, 831393, 1399655, 2346707, 3924873, 6541472, 10875575, 18025629, 29804125, 49143254, 80841455, 132651457, 217179366, 354745107, 578215807

Offset: 0

Author

Ilya Gutkovskiy, May 22 2018

Keywords

• nonn

Comments

Euler transform of A000065.

Convolution of the sequences A001970 and A010815.

Links

• N. J. A. Sloane, Transforms
• Index entries for sequences related to partitions

Crossrefs

Cf. A000041, A000065, A000219, A001383, A001970, A010815, A052847.

Programs

Maple

with(combinat): with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      numbpart(d)-d, d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, May 22 2018

Mathematica

nmax = 38; CoefficientList[Series[Product[1/(1 - x^k)^(PartitionsP[k] - 1), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (PartitionsP[d] - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 38}]

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^A000065(k).