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

Comments

In general, for m >= 1, if g.f.= Sum_{k>=1} x^(m*k^2)/Product_{j=1..m*k-1} (1-x^j), then a(n) ~ r^2 * (m*log(r)^2 + polylog(2, r^2))^(1/4) * exp(2*sqrt((m*log(r)^2 + polylog(2, r^2))*n)) / (2*sqrt(Pi*m*(m - (m-2)*r^2)) * n^(3/4)), where r is the positive real root of the equation r^2 = 1 - r^m. - Vaclav Kotesovec, Oct 14 2024

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 = 0.04778365700734... - Vaclav Kotesovec, Jan 26 2022

a(n) ~ r^2 * (5*log(r)^2 + polylog(2, r^2))^(1/4) * exp(2*sqrt((5*log(r)^2 + polylog(2, r^2))*n)) / (2*sqrt(5*Pi*(5 - 3*r^2)) * n^(3/4)), where r = 0.808730600479392... is the real root of the equation r^2 = 1 - r^5. - Vaclav Kotesovec, Oct 14 2024

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: A026810 A001400 A372703 * A008773 A008772 A351003

Adjacent sequences: A350894 A350895 A350896 * A350898 A350899 A350900

Keyword

nonn

Author

Seiichi Manyama, Jan 21 2022

Status

approved