A305840 Product_{n>=1} (1 + x^n)^a(n) = g.f. of A005169 (fountains of coins).
1, 1, 1, 2, 2, 4, 5, 10, 13, 23, 35, 59, 93, 154, 248, 413, 671, 1111, 1827, 3036, 5013, 8348, 13859, 23122, 38534, 64434, 107715, 180509, 302565, 508032, 853507, 1435828, 2416941, 4072943, 6868062, 11591918, 19577555, 33090308, 55964327, 94715248, 160391045
Offset: 1
Keywords
Examples
(1 + x) * (1 + x^2) * (1 + x^3) * (1 + x^4)^2 * (1 + x^5)^2 * ... * (1 + x^n)^a(n) * ... = 1/(1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - x^5/(1 - ...)))))).
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..3000
- N. J. A. Sloane, Transforms
- Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction
Programs
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Mathematica
nn = 39; f[x_] := Product[(1 + x^n)^a[n], {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1/(1 + ContinuedFractionK[-x^k, 1, {k, 1, nn}]), {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
Formula
Product_{n>=1} (1 + x^n)^a(n) = 1/(1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - x^5/(1 - ...)))))).
a(n) ~ 1 / (n * A347901^n). - Vaclav Kotesovec, Sep 18 2021
Comments