A363096 Number of partitions of n whose least part is a multiple of 5.

0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 1, 1, 1, 3, 2, 3, 3, 4, 7, 7, 8, 10, 11, 15, 16, 19, 22, 27, 34, 39, 46, 54, 63, 76, 86, 101, 117, 136, 161, 186, 214, 249, 287, 335, 384, 445, 509, 588, 677, 776, 888, 1020, 1163, 1334, 1519, 1735, 1975, 2253, 2564, 2917, 3312, 3762, 4265, 4842, 5477, 6203, 7012, 7928

Offset: 1

Author

Seiichi Manyama, May 19 2023

Keywords

• nonn

Comments

In general, for m > 0, if g.f. = Sum_{k>=1} x^(m*k)/Product_{j>=m*k} (1-x^j), then a(n) ~ Pi^(m-1) * (m-1)! * exp(Pi*sqrt(2*n/3)) / (2^(3/2) * 6^(m/2) * n^((m+1)/2)) * (1 - (m*(m+1)/(4*Pi) + (6*m^2 + 18*m + 1 + c)*Pi/144)/sqrt(n/6)), where c = 0 for m > 1 and c = -24 for m = 1. - Vaclav Kotesovec, May 21 2023

Links

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

Crossrefs

Cf. A026805, A363094, A363095.

Cf. A363047.

Programs

Mathematica

nmax = 60; Rest[CoefficientList[Series[Sum[x^(5*k)/QPochhammer[x^(5*k), x], {k, 1, nmax/5}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, May 20 2023 *)

PARI

my(N=70, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, N, x^(5*k)/prod(j=5*k, N, 1-x^j))))

Formula

G.f.: Sum_{k>=1} x^(5*k)/Product_{j>=5*k} (1-x^j).

a(n) ~ Pi^4 * exp(Pi*sqrt(2*n/3)) / (2*3^(3/2)*n^3) * (1 - (15*sqrt(6)/(2*Pi) + 241*Pi*sqrt(6)/144) / sqrt(n)). - Vaclav Kotesovec, May 21 2023