A344065 Expansion of Product_{k>=1} (1 + 6^(k-1)*x^k).

1, 1, 6, 42, 252, 1728, 10584, 71280, 435456, 2939328, 17962560, 119532672, 739031040, 4867527168, 30051689472, 198147658752, 1221537687552, 7984437608448, 49643697954816, 321998350270464, 1997815999463424, 12977575759282176, 80455233450737664, 519208351807832064

Offset: 0

Author

Ilya Gutkovskiy, May 08 2021

Keywords

• nonn

Crossrefs

Cf. A003056, A008289, A304961, A338675, A340103, A344062, A344063, A344064, A344066, A344067, A344068.

Programs

Mathematica

nmax = 23; CoefficientList[Series[Product[(1 + 6^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 6^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 23}]

PARI

seq(n)={Vec(prod(k=1, n, 1 + 6^(k-1)*x^k + O(x*x^n)))} \\ Andrew Howroyd, May 08 2021

Formula

a(n) = Sum_{k=0..A003056(n)} q(n,k) * 6^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.

a(n) ~ (-polylog(2, -1/6))^(1/4) * 6^n * exp(2*sqrt(-polylog(2, -1/6)*n)) / (2*sqrt(7*Pi/6)*n^(3/4)). - Vaclav Kotesovec, May 09 2021