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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.

%I A305536 #8 Jan 11 2023 12:31:12
%S A305536 1,1,3,12,62,410,3426,35360,438390,6358306,105544388,1970997142,
%T A305536 40860191470,930482058472,23079257369054,619157277351618,
%U A305536 17860295754328884,551188620179519302,18119420989759583998,632069815329176122584,23318435171385786420958,907077442499274638005314
%N 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.
%C A305536 Invert transform of A001515, shifted right one place.
%H A305536 Alois P. Heinz, <a href="/A305536/b305536.txt">Table of n, a(n) for n = 0..405</a>
%H A305536 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A305536 <a href="/index/Be#Bessel">Index entries for sequences related to Bessel functions or polynomials</a>
%F A305536 a(n) ~ 2^(n - 1/2) * n^(n-1) / exp(n-1). - _Vaclav Kotesovec_, Sep 18 2021
%p A305536 b:= proc(n) option remember;
%p A305536      `if`(n<2, n+1, (2*n-1)*b(n-1)+b(n-2))
%p A305536     end:
%p A305536 a:= proc(n) option remember;
%p A305536      `if`(n=0, 1, add(b(j-1)*a(n-j), j=1..n))
%p A305536     end:
%p A305536 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jan 11 2023
%t A305536 nmax = 21; CoefficientList[Series[1/(1 - x/(1 - x + ContinuedFractionK[-k x, 1 - x, {k, 1, nmax}])), {x, 0, nmax}], x]
%t A305536 nmax = 21; CoefficientList[Series[1/(1 - Sum[HypergeometricPFQ[{k, 1 - k}, {}, -1/2] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
%t A305536 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}]
%Y A305536 Cf. A001003, A001515, A144301, A258173.
%K A305536 nonn
%O A305536 0,3
%A A305536 _Ilya Gutkovskiy_, Jun 04 2018