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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.
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%I #7 Sep 18 2021 02:27:20

%S 1,1,3,13,75,557,5179,58589,784715,12154061,213593563,4195613373,

%T 91031201643,2160916171181,55687501548539,1547866851663261,

%U 46150908197995403,1469089501918434957,49722765216242122267,1782934051704982201469,67514992620138056010667

%N 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 Invert transform of A000165, shifted right one place.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

%F a(n) ~ 2^(n-1) * (n-1)!. - _Vaclav Kotesovec_, Sep 18 2021

%t nmax = 20; CoefficientList[Series[1/(1 - x/(1 + ContinuedFractionK[-2 Floor[(k + 1)/2] x, 1, {k, 1, nmax}])), {x, 0, nmax}], x]

%t nmax = 20; CoefficientList[Series[1/(1 - Sum[2^(k - 1) (k - 1)! x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

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

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jun 04 2018