A344068 - OEIS

%I #9 May 09 2021 03:35:22

%S 1,1,9,90,810,8019,72900,715149,6495390,63772920,579270690,5643903420,

%T 51613018479,499772430810,4567687565310,44250780833091,

%U 404188047763920,3894703308072990,35764052204589030,342923118899865390,3146016498406236720,30187757787717436380,276843069234653897241

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

%C 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

%F 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.

%F 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

%t nmax = 22; CoefficientList[Series[Product[(1 + 9^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]

%t Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 9^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 22}]

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

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

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, May 08 2021