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A305651 Expansion of Product_{k>=1} (1 + x^k)^(q(k)-1), where q(k) = number of partitions of k into distinct parts (A000009). 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified April 15 04:04 EDT 2021. Contains 342974 sequences. (Running on oeis4.)