A344066 Expansion of Product_{k>=1} (1 + 7^(k-1)*x^k).

1, 1, 7, 56, 392, 3087, 21952, 170471, 1210104, 9411920, 66824632, 513890832, 3683707839, 28086110472, 201122377288, 1534688027817, 10978118077136, 83158453503608, 599161640356888, 4508826988300152, 32435340235930576, 244366486039786096, 1756858874561956865, 13161303959340223232

Offset

0,3

Links

Table of n, a(n) for n=0..23.

Formula

a(n) = Sum_{k=0..A003056(n)} q(n,k) * 7^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.

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

Mathematica

nmax = 23; CoefficientList[Series[Product[(1 + 7^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 7^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 23}]

Prog

(PARI) seq(n)={Vec(prod(k=1, n, 1 + 7^(k-1)*x^k + O(x*x^n)))} \\ Andrew Howroyd, May 08 2021

Crossrefs

Cf. A003056, A008289, A304961, A338676, A340103, A344062, A344063, A344064, A344065, A344067, A344068.

Sequence in context: A264693 A054614 A270240 * A104896 A246939 A122996

Adjacent sequences: A344063 A344064 A344065 * A344067 A344068 A344069

Keyword

nonn

Author

Ilya Gutkovskiy, May 08 2021

Status

approved