A350898 - OEIS

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

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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 * A373076 A373070 A330878

Adjacent sequences: A350895 A350896 A350897 * A350899 A350900 A350901

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Jan 21 2022

STATUS

approved