A317911 Expansion of Product_{k>=2} 1/(1 - x^k)^p(k), where p(k) = number of partitions of k (A000041).

1, 0, 2, 3, 8, 13, 31, 53, 112, 201, 393, 710, 1343, 2409, 4431, 7912, 14255, 25223, 44787, 78519, 137700, 239347, 415343, 716001, 1231326, 2106287, 3593141, 6102679, 10335269, 17437476, 29337139, 49192762, 82261930, 137148782, 228061165, 378198633, 625623318, 1032301633

Offset: 0

Author

Ilya Gutkovskiy, Aug 10 2018

Keywords

• nonn

Comments

First differences of A001970.

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A000041, A001970, A002865, A191659, A291552.

Programs

Maple

with(numtheory): with(combinat):
b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      numbpart(d), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> b(n)-b(n-1):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 10 2018

Mathematica

nmax = 37; CoefficientList[Series[Product[1/(1 - x^k)^PartitionsP[k], {k, 2, nmax}], {x, 0, nmax}], x]
nmax = 37; CoefficientList[Series[Exp[Sum[Sum[PartitionsP[k] x^(j k)/j, {k, 2, nmax}], {j, 1, nmax}]], {x, 0, nmax}], x]
b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d PartitionsP[d], {d, Divisors[k]}] b[n - k], {k, 1, n}]/n]; Differences[Table[b[n], {n, -1, 37}]]

Formula

G.f.: exp(Sum_{j>=1} Sum_{k>=2} p(k)*x^(j*k)/j).