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A305840 Product_{n>=1} (1 + x^n)^a(n) = g.f. of A005169 (fountains of coins).

%I A305840 #11 Feb 16 2025 08:33:54
%S A305840 1,1,1,2,2,4,5,10,13,23,35,59,93,154,248,413,671,1111,1827,3036,5013,
%T A305840 8348,13859,23122,38534,64434,107715,180509,302565,508032,853507,
%U A305840 1435828,2416941,4072943,6868062,11591918,19577555,33090308,55964327,94715248,160391045
%N A305840 Product_{n>=1} (1 + x^n)^a(n) = g.f. of A005169 (fountains of coins).
%C A305840 Inverse weigh transform of A005169.
%H A305840 Alois P. Heinz, <a href="/A305840/b305840.txt">Table of n, a(n) for n = 1..3000</a>
%H A305840 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A305840 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html">Rogers-Ramanujan Continued Fraction</a>
%F A305840 Product_{n>=1} (1 + x^n)^a(n) = 1/(1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - x^5/(1 - ...)))))).
%F A305840 a(n) ~ 1 / (n * A347901^n). - _Vaclav Kotesovec_, Sep 18 2021
%e A305840 (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 - ...)))))).
%t A305840 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
%Y A305840 Cf. A005169, A226999.
%K A305840 nonn
%O A305840 1,4
%A A305840 _Ilya Gutkovskiy_, Jun 11 2018