A305651 Expansion of Product_{k>=1} (1 + x^k)^(q(k)-1), where q(k) = number of partitions of k into distinct parts (A000009).

1, 0, 0, 1, 1, 2, 3, 5, 7, 12, 17, 26, 39, 59, 87, 132, 192, 284, 419, 612, 892, 1303, 1887, 2730, 3945, 5677, 8154, 11689, 16711, 23839, 33960, 48244, 68432, 96888, 136922, 193148, 272058, 382508, 537007, 752735, 1053550, 1472406, 2054988, 2863993, 3986245, 5541008

Offset: 0

Author

Ilya Gutkovskiy, Jun 07 2018

Keywords

• nonn

Comments

Weigh transform of A111133.

Convolution of the sequences A050342 and A081362.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000009, A050342, A081362, A089259, A111133, A304966, A304969.

Programs

Maple

g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(g(i)-1, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 07 2018

Mathematica

nmax = 45; CoefficientList[Series[Product[(1 + x^k)^(PartitionsQ[k] - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[Exp[Sum[(-1)^(k + 1)/k (1/ QPochhammer[x^k, x^(2 k)] - 1/(1 - x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (PartitionsQ[d] - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 45}]

Formula

G.f.: Product_{k>=1} (1 + x^k)^A111133(k).

G.f.: Product_{k>=1} (1 + x^k)^(A000009(k)-1).