A304443 Coefficient of x^n in Product_{k>=1} (1+x^k)^(2*n).

1, 2, 10, 62, 394, 2562, 16966, 113794, 770442, 5254334, 36042250, 248403586, 1718732998, 11931569028, 83064794746, 579696375972, 4054279504266, 28408328186508, 199390547044342, 1401564307833908, 9865190079554954, 69522550703432476, 490484539061916794

Offset: 0

Author

Vaclav Kotesovec, May 12 2018

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A022567, A026011, A270913, A304444.

Programs

Mathematica

nmax = 25; Table[SeriesCoefficient[Product[(1+x^k)^(2*n), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
nmax = 25; Table[SeriesCoefficient[(QPochhammer[-1, x]/2)^(2*n), {x, 0, n}], {n, 0, nmax}]
(* Calculation of constants {d,c}: *) {1/r, Sqrt[Derivative[0, 1][QPochhammer][-1, r*s] / (Pi*r*(Sqrt[s]*Derivative[0, 1][QPochhammer][-1, r*s]^2 + 2*s*Derivative[0, 2][QPochhammer][-1, r*s]))]} /. FindRoot[{4*s == QPochhammer[-1, r*s]^2, 2*r*Sqrt[s]*Derivative[0, 1][QPochhammer][-1, r*s] == 2}, {r, 1/8}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Oct 03 2023 *)

Formula

a(n) ~ c * d^n / sqrt(n), where d = 7.21883059750200610514730564495768943165197819880185778427663522275469... and c = 0.300860732623379969554285615234449502629772950943717460278989499...