A305868 Product_{n>=1} 1/(1 - x^n)^a(n) = g.f. of A001147 (double factorial of odd numbers).

1, 2, 12, 87, 816, 9194, 122028, 1859460, 32002076, 613890984, 12989299596, 300556859080, 7550646317520, 204687481289946, 5955892982437120, 185158929516065160, 6125200081143892800, 214837724609502834082, 7963817560398871790604, 311101285877489780292000, 12773912991134665452205048

Offset

1,2

Comments

Inverse Euler transform of A001147.

Links

Seiichi Manyama, Table of n, a(n) for n = 1..404

N. J. A. Sloane, Transforms

Eric Weisstein's World of Mathematics, Double Factorial

Index entries for sequences related to factorial numbers

Formula

Product_{n>=1} 1/(1 - x^n)^a(n) = 1/(1 - x/(1 - 2*x/(1 - 3*x/(1 - 4*x/(1 - 5*x/(1 - ...)))))).

a(n) ~ 2^(n + 1/2) * n^n / exp(n). - Vaclav Kotesovec, Aug 09 2019

Example

1/((1 - x) * (1 - x^2)^2 * (1 - x^3)^12 * (1 - x^4)^87 * (1 - x^5)^816 * ... * (1 - x^n)^a(n) * ...) = 1 + 1*x + 1*3*x^2 + 1*3*5*x^3 + 1*3*5*7*x^4 + ... + A001147(k)*x^k + ...

Mathematica

nn = 21; f[x_] := Product[1/(1 - x^n)^a[n], {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1/(1 + ContinuedFractionK[-k x, 1, {k, 1, nn}]), {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
nmax = 20; s = ConstantArray[0, nmax]; Do[s[[j]] = j*(2*j - 1)!! - Sum[s[[d]]*(2*j - 2*d - 1)!!, {d, 1, j - 1}], {j, 1, nmax}]; Table[Sum[MoebiusMu[k/d]*s[[d]], {d, Divisors[k]}]/k, {k, 1, nmax}] (* Vaclav Kotesovec, Aug 09 2019 *)

Crossrefs

Cf. A001147, A112354, A305867, A305870.

Sequence in context: A290568 A181345 A193125 * A356830 A319324 A059435

Adjacent sequences: A305865 A305866 A305867 * A305869 A305870 A305871

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 12 2018

Status

approved