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

1,4

Comments

Inverse weigh transform of A005169.

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

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

Example

(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 - ...)))))).

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

Crossrefs

Cf. A005169, A226999.

Sequence in context: A195865 A222006 A127712 * A178113 A032090 A000014

Adjacent sequences: A305837 A305838 A305839 * A305841 A305842 A305843

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 11 2018

Status

approved