A305870 - OEIS

%I #9 Feb 16 2025 08:33:54

%S 1,3,12,90,816,9206,122028,1859550,32002076,613891800,12989299596,

%T 300556868286,7550646317520,204687481411974,5955892982437120,

%U 185158929517924710,6125200081143892800,214837724609534836158,7963817560398871790604,311101285877490394183800,12773912991134665452205048

%N Product_{n>=1} (1 + x^n)^a(n) = g.f. of A001147 (double factorial of odd numbers).

%C Inverse weigh transform of A001147.

%H Alois P. Heinz, <a href="/A305870/b305870.txt">Table of n, a(n) for n = 1..404</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DoubleFactorial.html">Double Factorial</a>

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

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

%e (1 + x) * (1 + x^2)^3 * (1 + x^3)^12 * (1 + x^4)^90 * (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 + ...

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p       add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))

%p     end:

%p a:= proc(n) option remember; doublefactorial(2*n-1)-b(n, n-1) end:

%p seq(a(n), n=1..23);  # _Alois P. Heinz_, Jun 13 2018

%t nn = 21; f[x_] := Product[(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

%Y Cf. A001147, A168246, A305868, A305869.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Jun 12 2018