A338675 Expansion of Product_{k>=1} 1 / (1 - 6^(k-1)*x^k).

1, 1, 7, 43, 295, 1807, 12391, 75895, 512647, 3179815, 21196807, 131258311, 875934727, 5416216711, 35763798535, 223059458311, 1461247179271, 9093600322567, 59586011601415, 370499158291975, 2411884242270727, 15072418547458567, 97530161503173127, 608700350537722375

Offset

0,3

Links

Table of n, a(n) for n=0..23.

Formula

a(n) = Sum_{k=0..n} p(n,k) * 6^(n-k), where p(n,k) = number of partitions of n into k parts.

a(n) ~ sqrt(5) * polylog(2, 1/6)^(1/4) * 6^(n - 1/2) * exp(2*sqrt(polylog(2, 1/6)*n)) / (2*sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

Mathematica

nmax = 23; CoefficientList[Series[Product[1/(1 - 6^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 6^(n - k), {k, 0, n}], {n, 0, 23}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 6^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 23}]

Crossrefs

Cf. A008284, A075900, A246938, A300579, A338673, A338674, A338676, A338677, A338678, A338679.

Sequence in context: A277188 A356559 A351757 * A244938 A199483 A331236

Adjacent sequences: A338672 A338673 A338674 * A338676 A338677 A338678

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 23 2021

Status

approved