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

1,10

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

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

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 *)

Prog

(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))))

Crossrefs

Cf. A026805, A363094, A363095.

Cf. A363047.

Sequence in context: A026536 A046213 A215625 * A260222 A181386 A193517

Adjacent sequences: A363093 A363094 A363095 * A363097 A363098 A363099

Keyword

nonn

Author

Seiichi Manyama, May 19 2023

Status

approved