A350893 Number of partitions of n such that (smallest part) = 2*(number of parts).

0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 10, 10, 12, 13, 15, 16, 19, 20, 23, 25, 28, 30, 34, 36, 40, 43, 47, 50, 56, 59, 65, 70, 77, 82, 91, 97, 107, 115, 126, 135, 149, 159, 174, 187, 204, 218, 238, 254, 276, 295, 320, 341, 370, 394, 426, 455, 491, 523, 565

Offset: 1

Author

Seiichi Manyama, Jan 21 2022

Keywords

• nonn

Crossrefs

Column 2 of A350890.

Cf. A168656, A237753, A237757.

Programs

Mathematica

nmax = 100; Rest[CoefficientList[1 + Series[Sum[x^(2*j^2)*(1 - x^j)/Product[1 - x^i, {i, 1, j}], {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jan 21 2022 *)

PARI

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

Formula

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

a(n) ~ (1 - alfa) * exp(2*sqrt(n*(2*log(alfa)^2 + polylog(2, 1 - alfa)))) * (2*log(alfa)^2 + polylog(2, 1 - alfa))^(1/4) / (2*sqrt(Pi) * sqrt(4 - 3*alfa) * n^(3/4)), where alfa = 0.72449195900051561158837228218703656578649448135... is positive real root of the equation alfa^4 + alfa - 1 = 0. - Vaclav Kotesovec, Jan 21 2022