A316230 Expansion of Product_{k>=1} 1/(1 + p(k)*x^k), where p(k) = number of partitions of k (A000041).

1, -1, -1, -2, 1, -2, 2, 0, 24, -17, 31, -21, 94, -107, 121, -443, 742, -977, 532, -2159, 3275, -6193, 6988, -11156, 30278, -39214, 42759, -80255, 149070, -193093, 291229, -451125, 1017812, -1335848, 1609412, -3248202, 5606551, -7684574, 10012531, -17908468

Offset

0,4

Links

Table of n, a(n) for n=0..39.

Eric Weisstein's World of Mathematics, Partition Function P

Index entries for sequences related to partitions

Formula

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

Mathematica

nmax = 39; CoefficientList[Series[Product[1/(1 + PartitionsP[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 39; CoefficientList[Series[Exp[Sum[Sum[(-1)^k PartitionsP[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (-PartitionsP[d])^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 39}]

Crossrefs

Cf. A000041, A063834, A271619, A304784, A304785, A316231.

Sequence in context: A108804 A284467 A200227 * A127249 A127251 A063251

Adjacent sequences: A316227 A316228 A316229 * A316231 A316232 A316233

Keyword

sign

Author

Ilya Gutkovskiy, Jun 27 2018

Status

approved