A304630 Expansion of (1/(1 - x))*Product_{k>=1} (1 - x^(3*k))/(1 - x^k).

1, 2, 4, 6, 10, 15, 22, 31, 44, 60, 82, 109, 145, 189, 246, 316, 405, 513, 648, 811, 1013, 1256, 1553, 1908, 2339, 2852, 3469, 4200, 5074, 6105, 7330, 8769, 10470, 12461, 14802, 17533, 20730, 24447, 28780, 33802, 39636, 46377, 54180, 63171, 73546, 85469, 99185, 114908, 132946, 153574, 177177

Offset

0,2

Comments

Partial sums of A000726.

Links

Table of n, a(n) for n=0..50.

Eric Weisstein's World of Mathematics, Partition Function b_k

Index entries for sequences related to partitions

Formula

G.f.: (1/(1 - x))*Product_{k>=0} 1/((1 - x^(3*k+1))*(1 - x^(3*k+2))).

G.f.: (1/(1 - x))*Product_{k>=1} (1 + x^k + x^(2*k)).

a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*Pi*n^(1/4)). - Vaclav Kotesovec, May 18 2018

Mathematica

nmax = 50; CoefficientList[Series[1/(1 - x) Product[(1 - x^(3 k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[1/(1 - x) Product[1/((1 - x^(3 k + 1)) (1 - x^(3 k + 2))), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[1/(1 - x) Product[(1 + x^k + x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A000070, A000726, A036469, A117276, A304632.

Sequence in context: A032417 A152415 A175826 * A073470 A086182 A035294

Adjacent sequences: A304627 A304628 A304629 * A304631 A304632 A304633

Keyword

nonn

Author

Ilya Gutkovskiy, May 15 2018

Status

approved