A286310 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 - ...))))).

1, 2, 14, 102, 944, 10378, 135134, 2026920, 34459410, 654728128, 13749310574, 316234132728, 7905853580624, 213458046541738, 6190283353628416, 191898783960483600, 6332659870762850624, 221643095476665302070, 8200794532637891559374, 319830986772877116086448

Offset

1,2

Links

Table of n, a(n) for n=1..20.

N. J. A. Sloane, Transforms

Formula

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

a(n) ~ 2^(n + 1/2) * n^n / exp(n). - Vaclav Kotesovec, Sep 16 2021

Example

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 - ...))))).

Mathematica

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

Crossrefs

Cf. A001147, A002996, A062794.

Sequence in context: A037621 A085372 A123525 * A295865 A355069 A378611

Adjacent sequences: A286307 A286308 A286309 * A286311 A286312 A286313

Keyword

nonn

Author

Ilya Gutkovskiy, May 06 2017

Status

approved