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

0,6

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

Formula

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

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

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}]

Crossrefs

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

Sequence in context: A326468 A326593 A123569 * A318185 A048816 A080528

Adjacent sequences: A305648 A305649 A305650 * A305652 A305653 A305654

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 07 2018

Status

approved