A338677 Expansion of Product_{k>=1} 1 / (1 - 8^(k-1)*x^k).

1, 1, 9, 73, 649, 5257, 46729, 378505, 3331721, 27219593, 237491849, 1938544265, 16925054601, 138041874057, 1196384310921, 9820024329865, 84609648809609, 693596417152649, 5977550934234761, 48976660041553545, 419984680697190025, 3455551232025810569, 29494747047731910281

Offset

0,3

Links

Table of n, a(n) for n=0..22.

Formula

a(n) = Sum_{k=0..n} p(n,k) * 8^(n-k), where p(n,k) = number of partitions of n into k parts.

a(n) ~ sqrt(7) * polylog(2, 1/8)^(1/4) * 8^(n - 1/2) * exp(2*sqrt(polylog(2, 1/8)*n)) / (2*sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

Mathematica

nmax = 22; CoefficientList[Series[Product[1/(1 - 8^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 8^(n - k), {k, 0, n}], {n, 0, 22}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 8^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 22}]

Crossrefs

Cf. A008284, A075900, A246940, A300579, A338673, A338674, A338675, A338676, A338678, A338679.

Sequence in context: A121246 A365774 A086226 * A199677 A197676 A367977

Adjacent sequences: A338674 A338675 A338676 * A338678 A338679 A338680

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 23 2021

Status

approved