A286310 - OEIS

%I #14 Nov 23 2024 03:33:57

%S 1,2,14,102,944,10378,135134,2026920,34459410,654728128,13749310574,

%T 316234132728,7905853580624,213458046541738,6190283353628416,

%U 191898783960483600,6332659870762850624,221643095476665302070,8200794532637891559374,319830986772877116086448

%N G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = 1/(1 - x/(1 - 2*x/(1 - 3*x/(1 - 4*x/(1 - ...))))).

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

%F Sum_{d|n} a(d) = A001147(n) for n > 0.

%F a(n) ~ 2^(n + 1/2) * n^n / exp(n). - _Vaclav Kotesovec_, Sep 16 2021

%e G.f.: 1 + x/(1 - x) + 2*x^2/(1 - x^2) + 14*x^3/(1 - x^3) + 102*x^4/(1 - x^4) + ... = 1/(1 - x/(1 - 2*x/(1 - 3*x/(1 - 4*x/(1 - ...))))).

%t nn = 20; f[x_] := 1 + Sum[a[n] x^n/(1 - x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1/(1 + ContinuedFractionK[-n x, 1, {n, 1, nn}]), {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

%t a[n_] := Sum[MoebiusMu[n/d] (2 d - 1)!!, {d, Divisors[n]}]; Array[a, 20]

%Y Cf. A001147, A002996, A062794.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, May 06 2017