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

Author

Ilya Gutkovskiy, Jun 04 2018

Keywords

• nonn

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

Crossrefs

Cf. A001003, A001515, A144301, A258173.

Programs

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}]

Formula

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