A350896 Number of partitions of n such that 4*(smallest part) = (number of parts).

0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 20, 22, 26, 30, 35, 40, 48, 55, 65, 76, 90, 105, 126, 147, 175, 206, 244, 286, 339, 396, 467, 545, 638, 741, 865, 1000, 1160, 1337, 1543, 1770, 2035, 2325, 2660, 3029, 3451, 3916, 4447, 5029, 5691, 6419, 7242, 8146, 9167, 10286, 11546, 12930, 14481, 16185

Offset

1,6

Links

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

Formula

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

a(n) ~ c * exp(Pi*sqrt(2*n/5)) / n^(3/4), where c = (3 - sqrt(5))^(1/4) / (8*sqrt(5)) = 0.05226232058... - Vaclav Kotesovec, Jan 25 2022, updated Oct 13 2024

Example

For n=7 there are a(7)=3 such partitions: [1,2,2,2], [1,1,2,3] and [1,1,1,4]. - R. J. Mathar, Jun 20 2022

Mathematica

CoefficientList[Series[Sum[x^(4k^2)/Product[1-x^j, {j, 4k-1}], {k, 63}], {x, 0, 63}], x] (* Stefano Spezia, Jan 22 2022 *)

Prog

(PARI) my(N=66, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, sqrtint(N\4), x^(4*k^2)/prod(j=1, 4*k-1, 1-x^j))))

Crossrefs

Column 4 of A350889.

Cf. A168657.

Sequence in context: A008755 A029006 A085756 * A008754 A029005 A132154

Adjacent sequences: A350893 A350894 A350895 * A350897 A350898 A350899

Keyword

nonn

Author

Seiichi Manyama, Jan 21 2022

Status

approved