A344068 Expansion of Product_{k>=1} (1 + 9^(k-1)*x^k).

1, 1, 9, 90, 810, 8019, 72900, 715149, 6495390, 63772920, 579270690, 5643903420, 51613018479, 499772430810, 4567687565310, 44250780833091, 404188047763920, 3894703308072990, 35764052204589030, 342923118899865390, 3146016498406236720, 30187757787717436380, 276843069234653897241

Offset

0,3

Comments

In general, if g.f. = Product_{k>=1} (1 + d^(k-1)*x^k), where d > 1, then a(n) ~ (-polylog(2, -1/d))^(1/4) * d^n * exp(2*sqrt(-polylog(2, -1/d)*n)) / (2*sqrt((1 + 1/d)*Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

Links

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

Formula

a(n) = Sum_{k=0..A003056(n)} q(n,k) * 9^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.

a(n) ~ (-polylog(2, -1/9))^(1/4) * 9^n * exp(2*sqrt(-polylog(2, -1/9)*n)) / (2*sqrt(10*Pi/9)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

Mathematica

nmax = 22; CoefficientList[Series[Product[(1 + 9^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 9^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 22}]

Prog

(PARI) seq(n)={Vec(prod(k=1, n, 1 + 9^(k-1)*x^k + O(x*x^n)))} \\ Andrew Howroyd, May 08 2021

Crossrefs

Cf. A003056, A008289, A304961, A338678, A340103, A344062, A344063, A344064, A344065, A344066, A344067.

Sequence in context: A261315 A270242 A054616 * A052386 A246941 A186510

Adjacent sequences: A344065 A344066 A344067 * A344069 A344070 A344071

Keyword

nonn

Author

Ilya Gutkovskiy, May 08 2021

Status

approved