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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.
0
1, 1, 3, 13, 75, 557, 5179, 58589, 784715, 12154061, 213593563, 4195613373, 91031201643, 2160916171181, 55687501548539, 1547866851663261, 46150908197995403, 1469089501918434957, 49722765216242122267, 1782934051704982201469, 67514992620138056010667
OFFSET
0,3
COMMENTS
Invert transform of A000165, shifted right one place.
FORMULA
a(n) ~ 2^(n-1) * (n-1)!. - Vaclav Kotesovec, Sep 18 2021
MATHEMATICA
nmax = 20; CoefficientList[Series[1/(1 - x/(1 + ContinuedFractionK[-2 Floor[(k + 1)/2] x, 1, {k, 1, nmax}])), {x, 0, nmax}], x]
nmax = 20; CoefficientList[Series[1/(1 - Sum[2^(k - 1) (k - 1)! x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jun 04 2018
STATUS
approved