A338676 Expansion of Product_{k>=1} 1 / (1 - 7^(k-1)*x^k).

1, 1, 8, 57, 449, 3193, 25145, 178809, 1391314, 9996498, 76955586, 552257546, 4255024523, 30502987019, 232969386483, 1682476714724, 12762937304013, 92019035596293, 698222541789109, 5030814634614406, 37955614705675479, 274741644961416648, 2061916926761604144, 14909943849253537057

Offset

0,3

Links

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

Formula

a(n) = Sum_{k=0..n} p(n,k) * 7^(n-k), where p(n,k) = number of partitions of n into k parts.

a(n) ~ sqrt(6) * polylog(2, 1/7)^(1/4) * 7^(n - 1/2) * exp(2*sqrt(polylog(2, 1/7)*n)) / (2*sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

Mathematica

nmax = 23; CoefficientList[Series[Product[1/(1 - 7^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 7^(n - k), {k, 0, n}], {n, 0, 23}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 7^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 23}]

Crossrefs

Cf. A008284, A075900, A246939, A300579, A338673, A338674, A338675, A338677, A338678, A338679.

Sequence in context: A281912 A343352 A307642 * A199555 A241594 A241595

Adjacent sequences: A338673 A338674 A338675 * A338677 A338678 A338679

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 23 2021

Status

approved