A338678 Expansion of Product_{k>=1} 1 / (1 - 9^(k-1)*x^k).

1, 1, 10, 91, 901, 8191, 81091, 737191, 7239142, 66288142, 646149322, 5912729632, 57664985653, 527352541453, 5111015223223, 46998961540624, 453182267869615, 4163124744738505, 40151590267580785, 368699990679135946, 3540322181970716707, 32632895079429817528, 312061810101214595698

Offset: 0

Author

Ilya Gutkovskiy, Apr 23 2021

Keywords

• nonn

Crossrefs

Cf. A008284, A075900, A246941, A300579, A338673, A338674, A338675, A338676, A338677, A338679.

Programs

Mathematica

nmax = 22; CoefficientList[Series[Product[1/(1 - 9^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 9^(n - k), {k, 0, n}], {n, 0, 22}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 9^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 22}]

Formula

a(n) = Sum_{k=0..n} p(n,k) * 9^(n-k), where p(n,k) = number of partitions of n into k parts.

a(n) ~ sqrt(2) * polylog(2, 1/9)^(1/4) * 9^(n - 1/2) * exp(2*sqrt(polylog(2, 1/9)*n)) / (sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021