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 A307495 Expansion of Sum_{k>=0} k!*((1 - sqrt(1 - 4*x))/2)^k. 0

%I

%S 1,1,3,12,57,312,1950,13848,111069,998064,9957186,109305240,

%T 1309637274,17006109072,237888664572,3566114897520,57030565449765,

%U 969154436550240,17439499379433690,331268545604793240,6624013560942038670,139080391965533653200,3059323407592802838180,70355685298375014175440

%N Expansion of Sum_{k>=0} k!*((1 - sqrt(1 - 4*x))/2)^k.

%C Catalan transform of A000142 (factorial numbers).

%F G.f.: 1 /(1 - x*c(x)/(1 - x*c(x)/(1 - 2*x*c(x)/(1 - 2*x*c(x)/(1 - 3*x*c(x)/(1 - 3*x*c(x)/(1 - ...))))))), a continued fraction, where c(x) = g.f. of Catalan numbers (A000108).

%F Sum_{n>=0} a(n)*(x*(1 - x))^n = g.f. of A000142.

%F a(n) = (1/n) * Sum_{k=1..n} binomial(2*n-k-1,n-k)*k*k! for n > 0.

%F a(n) ~ exp(1) * n!. - _Vaclav Kotesovec_, Aug 10 2019

%t nmax = 23; CoefficientList[Series[Sum[k! ((1 - Sqrt[1 - 4 x])/2)^k, {k, 0, nmax}], {x, 0, nmax}], x]

%t nmax = 23; CoefficientList[Series[1/(1 + ContinuedFractionK[-Floor[(k + 1)/2] (1 - Sqrt[1 - 4 x])/2, 1, {k, 1, nmax}]), {x, 0, nmax}], x]

%t Join[{1}, Table[1/n Sum[Binomial[2n - k - 1, n - k] k k!, {k, n}], {n, 23}]]

%Y Cf. A000108, A000142, A013999, A100100.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Apr 10 2019

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Last modified December 5 23:35 EST 2019. Contains 329783 sequences. (Running on oeis4.)