A344064 Expansion of Product_{k>=1} (1 + 5^(k-1)*x^k).

1, 1, 5, 30, 150, 875, 4500, 25625, 131250, 750000, 3843750, 21562500, 112109375, 621093750, 3222656250, 17880859375, 92578125000, 508300781250, 2658691406250, 14465332031250, 75439453125000, 411254882812500, 2142486572265625, 11590576171875000, 60722351074218750, 326728820800781250

Offset: 0

Author

Ilya Gutkovskiy, May 08 2021

Keywords

• nonn

Crossrefs

Cf. A003056, A008289, A261569, A304961, A338674, A340103, A344062, A344063, A344065, A344066, A344067, A344068.

Programs

Mathematica

nmax = 25; CoefficientList[Series[Product[(1 + 5^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 5^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 25}]

PARI

seq(n)={Vec(prod(k=1, n, 1 + 5^(k-1)*x^k + O(x*x^n)))} \\ Andrew Howroyd, May 08 2021

Formula

a(n) = Sum_{k=0..A003056(n)} q(n,k) * 5^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.

a(n) ~ (-polylog(2, -1/5))^(1/4) * 5^n * exp(2*sqrt(-polylog(2, -1/5)*n)) / (2*sqrt(6*Pi/5)*n^(3/4)). - Vaclav Kotesovec, May 09 2021