A350899 Number of partitions of n such that (smallest part) = 5*(number of parts).

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 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, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 20, 21, 22, 24, 25, 27, 29, 31, 33, 36, 38, 41, 44, 47

Offset

1,45

Links

Table of n, a(n) for n=1..93.

Formula

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

a(n) ~ (1 - alfa) * exp(2*sqrt(n*(5*log(alfa)^2 + polylog(2, 1 - alfa)))) * (5*log(alfa)^2 + polylog(2, 1 - alfa))^(1/4) / (2*sqrt(Pi) * sqrt(10 - 9*alfa) * n^(3/4)), where alfa = 0.8350790427235590476091499923248865165628469558282... is positive real root of the equation alfa^10 + alfa - 1 = 0. - Vaclav Kotesovec, Jan 22 2022

Prog

(PARI) my(N=99, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, sqrtint(N\5), x^(5*k^2)/prod(j=1, k-1, 1-x^j))))

Crossrefs

Column 5 of A350890.

Cf. A168656.

Sequence in context: A123108 A008619 A110654 * A350898 A330878 A350894

Adjacent sequences: A350896 A350897 A350898 * A350900 A350901 A350902

Keyword

nonn

Author

Seiichi Manyama, Jan 21 2022

Status

approved