A350897 Number of partitions of n such that 5*(smallest part) = (number of parts).

0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 27, 34, 39, 47, 55, 65, 74, 87, 99, 115, 131, 151, 172, 199, 226, 260, 298, 343, 393, 454, 522, 603, 696, 804, 929, 1076, 1243, 1438, 1664, 1924, 2222, 2567, 2961, 3413, 3931, 4520, 5193, 5959, 6827, 7811, 8928, 10186, 11607, 13208, 15008, 17028, 19297

Offset

1,7

Links

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

Formula

G.f.: Sum_{k>=1} x^(5*k^2)/Product_{j=1..5*k-1} (1-x^j).

a(n) ~ c * exp(Pi*sqrt(r*n)) / n^(3/4), where r = 0.42067169741517... and c = 0.04778365700737... - Vaclav Kotesovec, Jan 26 2022

Mathematica

CoefficientList[Series[Sum[x^(5k^2)/Product[1-x^j, {j, 5k-1}], {k, 62}], {x, 0, 62}], x] (* Stefano Spezia, Jan 22 2022 *)

Prog

(PARI) my(N=66, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, sqrtint(N\5), x^(5*k^2)/prod(j=1, 5*k-1, 1-x^j))))

Crossrefs

Column 5 of A350889.

Cf. A168657.

Sequence in context: A242717 A026810 A001400 * A008773 A008772 A351003

Adjacent sequences: A350894 A350895 A350896 * A350898 A350899 A350900

Keyword

nonn

Author

Seiichi Manyama, Jan 21 2022

Status

approved