A363095 Number of partitions of n whose least part is a multiple of 4.

0, 0, 0, 1, 0, 0, 0, 2, 1, 1, 1, 3, 2, 3, 3, 7, 6, 8, 9, 13, 13, 17, 19, 28, 30, 38, 43, 56, 62, 76, 87, 110, 124, 151, 173, 211, 241, 289, 332, 399, 456, 539, 620, 733, 838, 983, 1127, 1322, 1513, 1761, 2016, 2343, 2677, 3096, 3536, 4083, 4655, 5355, 6101, 7005, 7969, 9124, 10370, 11856, 13453, 15340

Offset

1,8

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Formula

G.f.: Sum_{k>=1} x^(4*k)/Product_{j>=4*k} (1-x^j).

a(n) ~ Pi^3 * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(5/2)) * (1 - (5*sqrt(6)/Pi + 169*Pi*sqrt(6)/144)/sqrt(n)). - Vaclav Kotesovec, May 21 2023

Mathematica

nmax = 60; Rest[CoefficientList[Series[Sum[x^(4*k)/QPochhammer[x^(4*k), x], {k, 1, nmax/4}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, May 20 2023 *)

Prog

(PARI) my(N=70, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/prod(j=4*k, N, 1-x^j))))

Crossrefs

Cf. A026805, A363094, A363096.

Cf. A363046.

Sequence in context: A290601 A344446 A094899 * A057774 A089355 A302126

Adjacent sequences: A363092 A363093 A363094 * A363096 A363097 A363098

Keyword

nonn

Author

Seiichi Manyama, May 19 2023

Status

approved