A305867 Expansion of Product_{k>=1} 1/(1 - x^k)^(2*k-1)!!.

1, 1, 4, 19, 130, 1120, 11960, 151595, 2230550, 37361755, 701873371, 14610774346, 333746628499, 8298025724194, 223049950124065, 6444634486214748, 199165237980655863, 6555102341516877027, 228905611339161301812, 8452656930719845696590, 329075775511339959533232, 13471099892869946627980017

Offset: 0

Author

Ilya Gutkovskiy, Jun 12 2018

Keywords

• nonn

Comments

Euler transform of A001147.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..404
• N. J. A. Sloane, Transforms
• Eric Weisstein's World of Mathematics, Double Factorial
• Index entries for sequences related to factorial numbers

Crossrefs

Cf. A001147, A107895, A179327, A261047, A280088, A305868, A305869.

Programs

Maple

N:= 25:
S:=series(mul((1-x^k)^(-doublefactorial(2*k-1)),k=1..N),x,N+1):
seq(coeff(S,x,n),n=0..N); # Robert Israel, Jun 12 2018

Mathematica

nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(2 k - 1)!!, {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (2 d - 1)!!, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 21}]

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^A001147(k).