A303071 a(n) = [x^n] (1/(1 - x))*Product_{k>=1} (1 + x^k)^n.

1, 2, 6, 23, 90, 362, 1491, 6225, 26242, 111479, 476466, 2046464, 8825559, 38191467, 165751529, 721177328, 3144703234, 13739010855, 60127642329, 263545670385, 1156732481150, 5083320593976, 22364017244278, 98491038664903, 434160710647831, 1915482295831037, 8457663096970431

Offset: 0

Author

Ilya Gutkovskiy, Apr 18 2018

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..300
• Index entries for sequences related to partitions

Crossrefs

Cf. A036469, A270913, A286335, A303070.

Programs

Mathematica

Table[SeriesCoefficient[1/(1 - x) Product[(1 + x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[1/(1 - x) Exp[n Sum[(-1)^(k + 1) x^k/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 26}]

Formula

a(n) = [x^n] (1/(1 - x))*exp(n*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))).

a(n) = Sum_{j=0..n} A286335(j,n).

a(n) ~ c * d^n / sqrt(n), where d = A270914 = 4.5024767476173544877385939327007... and c = 0.44252758868364961050787771300805... - Vaclav Kotesovec, May 19 2018