A344062 Expansion of Product_{k>=1} (1 + 3^(k-1)*x^k).

1, 1, 3, 12, 36, 135, 432, 1539, 4860, 17496, 55404, 192456, 623295, 2125764, 6849684, 23442453, 75110328, 252965916, 822670668, 2735858268, 8838926712, 29501352792, 95090206689, 314068876416, 1018141045092, 3342663979092, 10798571289897, 35481518064576

Offset: 0

Author

Ilya Gutkovskiy, May 08 2021

Keywords

• nonn

Crossrefs

Cf. A003056, A008289, A032308, A300579, A304961, A340103, A344063, A344064, A344065, A344066, A344067, A344068.

Programs

Mathematica

nmax = 27; CoefficientList[Series[Product[(1 + 3^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 3^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 27}]

PARI

seq(n)={Vec(prod(k=1, n, 1 + 3^(k-1)*x^k + O(x*x^n)))} \\ Andrew Howroyd, May 08 2021

Formula

a(n) = Sum_{k=0..A003056(n)} q(n,k) * 3^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.

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