This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A305535 #7 Sep 18 2021 02:27:20
%S A305535 1,1,3,13,75,557,5179,58589,784715,12154061,213593563,4195613373,
%T A305535 91031201643,2160916171181,55687501548539,1547866851663261,
%U A305535 46150908197995403,1469089501918434957,49722765216242122267,1782934051704982201469,67514992620138056010667
%N 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.
%C A305535 Invert transform of A000165, shifted right one place.
%H A305535 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A305535 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A305535 a(n) ~ 2^(n-1) * (n-1)!. - _Vaclav Kotesovec_, Sep 18 2021
%t A305535 nmax = 20; CoefficientList[Series[1/(1 - x/(1 + ContinuedFractionK[-2 Floor[(k + 1)/2] x, 1, {k, 1, nmax}])), {x, 0, nmax}], x]
%t A305535 nmax = 20; CoefficientList[Series[1/(1 - Sum[2^(k - 1) (k - 1)! x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
%t A305535 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}]
%Y A305535 Cf. A000165, A051295, A051296, A112934, A141307, A292778.
%K A305535 nonn
%O A305535 0,3
%A A305535 _Ilya Gutkovskiy_, Jun 04 2018