A305535 Expansion of 1/(1 - x/(1 - 2*x/(1 - 2*x/(1 - 4*x/(1 - 4*x/(1 - 6*x/(1 - 6*x/(1 - ...)))))))), a continued fraction.

1, 1, 3, 13, 75, 557, 5179, 58589, 784715, 12154061, 213593563, 4195613373, 91031201643, 2160916171181, 55687501548539, 1547866851663261, 46150908197995403, 1469089501918434957, 49722765216242122267, 1782934051704982201469, 67514992620138056010667

Offset

0,3

Comments

Invert transform of A000165, shifted right one place.

Links

Table of n, a(n) for n=0..20.

N. J. A. Sloane, Transforms

Index entries for sequences related to factorial numbers

Formula

a(n) ~ 2^(n-1) * (n-1)!. - Vaclav Kotesovec, Sep 18 2021

O.g.f. satisfies A(x) = (1-x)*A(x)^2 - 2*x^2*A'(x). - Paul D. Hanna, Mar 29 2026

Example

G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 75*x^4 + 557*x^5 + 5179*x^6 + 58589*x^7 + 784715*x^8 + ...

Mathematica

nmax = 20; CoefficientList[Series[1/(1 - x/(1 + ContinuedFractionK[-2 Floor[(k + 1)/2] x, 1, {k, 1, nmax}])), {x, 0, nmax}], x]
nmax = 20; CoefficientList[Series[1/(1 - Sum[2^(k - 1) (k - 1)! x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[2^(k - 1) (k - 1)! a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

Crossrefs

Cf. A000165, A051295, A051296, A112934, A141307, A292778.

Sequence in context: A034172 A000670 A032036 * A300793 A222427 A026072

Adjacent sequences: A305532 A305533 A305534 * A305536 A305537 A305538

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 04 2018

Status

approved