A350898 Number of partitions of n such that (smallest part) = 4*(number of parts).

0, 0, 0, 1, 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, 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, 17, 17, 19, 20, 22, 23, 26, 27, 30, 32, 35, 37, 41, 43, 47, 50, 54, 57, 62, 65, 70, 74, 79, 83, 89, 93, 99, 104

Offset

1,36

Links

Table of n, a(n) for n=1..91.

Formula

G.f.: Sum_{k>=1} x^(4*k^2)/Product_{j=1..k-1} (1-x^j).

a(n) ~ (1 - alfa) * exp(2*sqrt(n*(4*log(alfa)^2 + polylog(2, 1 - alfa)))) * (4*log(alfa)^2 + polylog(2, 1 - alfa))^(1/4) / (2*sqrt(Pi) * sqrt(8 - 7*alfa) * n^(3/4)), where alfa = 0.8116523200278026483934188589034567041719182934245... is positive real root of the equation alfa^8 + alfa - 1 = 0. - Vaclav Kotesovec, Jan 22 2022

Prog

(PARI) my(N=99, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, sqrtint(N\4), x^(4*k^2)/prod(j=1, k-1, 1-x^j))))

Crossrefs

Column 4 of A350890.

Cf. A168656.

Sequence in context: A008619 A110654 A350899 * A330878 A350894 A109728

Adjacent sequences: A350895 A350896 A350897 * A350899 A350900 A350901

Keyword

nonn

Author

Seiichi Manyama, Jan 21 2022

Status

approved