A304444 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^(2*n).

1, 2, 14, 98, 726, 5512, 42614, 333608, 2636326, 20985272, 168012824, 1351507830, 10914317934, 88432329546, 718545161208, 5852747363518, 47774241056710, 390702055798978, 3200542803221192, 26257321971526646, 215705170816632376, 1774181109262878848

Offset

0,2

Links

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

Formula

a(n) ~ c * d^n / sqrt(n), where d = 8.42516721063251541777601555584151410936132980324698494327338254953123205... and c = 0.29923152009652750283923119244187982714171590056794904644563876...

Mathematica

nmax = 25; Table[SeriesCoefficient[Product[1/(1-x^k)^(2*n), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
nmax = 25; Table[SeriesCoefficient[1/QPochhammer[x]^(2*n), {x, 0, n}], {n, 0, nmax}]
(* Calculation of constants {d, c}: *) eq = FindRoot[{1/QPochhammer[r*s]^2 == s, 1/s + 2*r*Sqrt[s]*Derivative[0, 1][QPochhammer][r*s, r*s] == (2*(Log[1 - r*s] + QPolyGamma[0, 1, r*s]))/(s* Log[r*s])}, {r, 1/8}, {s, 2}, WorkingPrecision -> 1000]; {N[1/r /. eq, 120], val = Sqrt[((1 - r*s)*Log[r*s]^2)/(Pi*(16*r*s*ArcTanh[1 - 2*r*s] - (-1 + r*s)*(Log[r*s] - 2*Log[1 - r*s])*(3*Log[r*s] - 2*Log[1 - r*s]) - 8*Log[1 - r*s] - 8*(-1 + r*s)*(-1 + 2*ArcTanh[1 - 2*r*s])* QPolyGamma[0, 1, r*s] + (4 - 4*r*s)* QPolyGamma[0, 1, r*s]^2 + 4*(-1 + r*s)*(QPolyGamma[1, 1, r*s] + r*s*Log[r*s] * (r*s^(3/2)*Log[r*s]* Derivative[0, 2][QPochhammer][r*s, r*s] - 2*Derivative[0, 0, 1][QPolyGamma][0, 1, r*s]))))] /. eq; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3]} (* Vaclav Kotesovec, Oct 03 2023 *)

Crossrefs

Cf. A000712, A008485, A161870, A171803, A304443.

Sequence in context: A286445 A322262 A109808 * A370617 A247481 A037516

Adjacent sequences: A304441 A304442 A304443 * A304445 A304446 A304447

Keyword

nonn

Author

Vaclav Kotesovec, May 12 2018

Status

approved