A344067 Expansion of Product_{k>=1} (1 + 8^(k-1)*x^k).

1, 1, 8, 72, 576, 5120, 41472, 364544, 2949120, 25952256, 209977344, 1830813696, 14931722240, 129251672064, 1053340729344, 9123584278528, 74294344286208, 639503450505216, 5239722662166528, 44846880273727488, 367008185258606592, 3144110674230116352, 25718087147075928064

Offset

0,3

Links

Table of n, a(n) for n=0..22.

Formula

a(n) = Sum_{k=0..A003056(n)} q(n,k) * 8^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.

a(n) ~ (-polylog(2, -1/8))^(1/4) * 8^n * exp(2*sqrt(-polylog(2, -1/8)*n)) / (6*sqrt(Pi/8)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

Mathematica

nmax = 22; CoefficientList[Series[Product[(1 + 8^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 8^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 22}]

Prog

(PARI) seq(n)={Vec(prod(k=1, n, 1 + 8^(k-1)*x^k + O(x*x^n)))} \\ Andrew Howroyd, May 08 2021

Crossrefs

Cf. A003056, A008289, A304961, A338677, A340103, A344062, A344063, A344064, A344065, A344066, A344068.

Sequence in context: A304826 A270241 A054615 * A111919 A052379 A246940

Adjacent sequences: A344064 A344065 A344066 * A344068 A344069 A344070

Keyword

nonn

Author

Ilya Gutkovskiy, May 08 2021

Status

approved