A350894 Number of partitions of n such that (smallest part) = 3*(number of parts).

0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 13, 14, 15, 17, 18, 20, 22, 24, 26, 29, 31, 34, 37, 40, 43, 47, 50, 54, 58, 62, 66, 71, 75, 80, 85, 90, 95, 102, 107, 114, 121, 129, 136, 146, 154, 165, 175, 187, 198, 213

Offset: 1

Author

Seiichi Manyama, Jan 21 2022

Keywords

• nonn

Links

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

Crossrefs

Column 3 of A350890.

Cf. A168656, A237756.

Programs

PARI

my(N=99, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, sqrtint(N\3), x^(3*k^2)/prod(j=1, k-1, 1-x^j))))

Formula

G.f.: Sum_{k>=1} x^(3*k^2)/Product_{j=1..k-1} (1-x^j).

a(n) ~ (1 - alfa) * exp(2*sqrt(n*(3*log(alfa)^2 + polylog(2, 1 - alfa)))) * (3*log(alfa)^2 + polylog(2, 1 - alfa))^(1/4) / (2*sqrt(Pi) * sqrt(6 - 5*alfa) * n^(3/4)), where alfa = 0.7780895986786010978806823096592944458720784440255... is positive real root of the equation alfa^6 + alfa - 1 = 0. - Vaclav Kotesovec, Jan 22 2022