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A344068
Expansion of Product_{k>=1} (1 + 9^(k-1)*x^k).
6
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
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
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, May 08 2021
STATUS
approved