A363094 Number of partitions of n whose least part is a multiple of 3.

0, 0, 1, 0, 0, 2, 1, 1, 3, 2, 3, 6, 6, 7, 11, 11, 14, 21, 24, 29, 38, 44, 54, 69, 81, 98, 123, 144, 174, 213, 253, 300, 363, 427, 508, 608, 716, 846, 1004, 1176, 1384, 1631, 1908, 2230, 2616, 3046, 3553, 4143, 4813, 5586, 6492, 7509, 8693, 10057, 11608, 13383, 15435, 17753, 20418, 23463, 26923, 30864

Offset

1,6

Links

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

Formula

G.f.: Sum_{k>=1} x^(3*k)/Product_{j>=3*k} (1-x^j).

a(n) ~ Pi^2 * exp(Pi*sqrt(2*n/3)) / (4 * 3^(3/2) * n^2) * (1 - (3*sqrt(6)/Pi + 109*Pi*sqrt(6)/144)/sqrt(n)). - Vaclav Kotesovec, May 21 2023

Mathematica

nmax = 60; Rest[CoefficientList[Series[Sum[x^(3*k)/QPochhammer[x^(3*k), x], {k, 1, nmax/3}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, May 20 2023 *)

Prog

(PARI) my(N=70, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(3*k)/prod(j=3*k, N, 1-x^j))))

Crossrefs

Cf. A026805, A363095, A363096.

Cf. A363045.

Sequence in context: A366398 A365932 A240853 * A319849 A380247 A384772

Adjacent sequences: A363091 A363092 A363093 * A363095 A363096 A363097

Keyword

nonn

Author

Seiichi Manyama, May 19 2023

Status

approved