A305536 Expansion of 1/(1 - x/(1 - x - 1*x/(1 - x - 2*x/(1 - x - 3*x/(1 - x - 4*x/(1 - ...)))))), a continued fraction.

1, 1, 3, 12, 62, 410, 3426, 35360, 438390, 6358306, 105544388, 1970997142, 40860191470, 930482058472, 23079257369054, 619157277351618, 17860295754328884, 551188620179519302, 18119420989759583998, 632069815329176122584, 23318435171385786420958, 907077442499274638005314

Offset

0,3

Comments

Invert transform of A001515, shifted right one place.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..405

N. J. A. Sloane, Transforms

Index entries for sequences related to Bessel functions or polynomials

Formula

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

Maple

b:= proc(n) option remember;
     `if`(n<2, n+1, (2*n-1)*b(n-1)+b(n-2))
    end:
a:= proc(n) option remember;
     `if`(n=0, 1, add(b(j-1)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 11 2023

Mathematica

nmax = 21; CoefficientList[Series[1/(1 - x/(1 - x + ContinuedFractionK[-k x, 1 - x, {k, 1, nmax}])), {x, 0, nmax}], x]
nmax = 21; CoefficientList[Series[1/(1 - Sum[HypergeometricPFQ[{k, 1 - k}, {}, -1/2] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[HypergeometricPFQ[{k, 1 - k}, {}, -1/2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]

Crossrefs

Cf. A001003, A001515, A144301, A258173.

Sequence in context: A020033 A266329 A208734 * A121123 A361882 A020123

Adjacent sequences: A305533 A305534 A305535 * A305537 A305538 A305539

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 04 2018

Status

approved