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

Author

Seiichi Manyama, Jan 21 2022

Keywords

• nonn

Examples

			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
		

Links

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

Crossrefs

Column 4 of A350889.

Cf. A168657.

Programs

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 *)

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))))

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