A305869 Expansion of Product_{k>=1} (1 + x^k)^(2*k-1)!!.

1, 1, 3, 18, 123, 1098, 11806, 150406, 2218065, 37206485, 699604235, 14572941915, 333037896380, 8283300923765, 222714069807495, 6436292165450693, 198941178161054798, 6548632634238445779, 228705772883364303114, 8446082393596031365629, 328846269698068735291665, 13462627492562640070346824

Offset: 0

Author

Ilya Gutkovskiy, Jun 12 2018

Keywords

• nonn

Comments

Weigh transform of A001147.

Links

• Alois P. Heinz, 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, A261052, A305867, A305870.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(doublefactorial(2*i-1), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 13 2018

Mathematica

nmax = 21; CoefficientList[Series[Product[(1 + x^k)^(2 k - 1)!!, {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) 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 + x^k)^A001147(k).