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

%I A307495 #13 Jan 28 2020 12:13:11
%S A307495 1,1,3,12,57,312,1950,13848,111069,998064,9957186,109305240,
%T A307495 1309637274,17006109072,237888664572,3566114897520,57030565449765,
%U A307495 969154436550240,17439499379433690,331268545604793240,6624013560942038670,139080391965533653200,3059323407592802838180,70355685298375014175440
%N A307495 Expansion of Sum_{k>=0} k!*((1 - sqrt(1 - 4*x))/2)^k.
%C A307495 Catalan transform of A000142 (factorial numbers).
%C A307495 From _Peter Bala_, Jan 27 2020: (Start)
%C A307495 This sequence is the main diagonal of the lower triangular array formed by putting the sequence of factorial numbers in the first column (k = 0) of the array and then completing the triangle using the relation T(n,k) = T(n-1,k) + T(n,k-1) for k >= 1.
%C A307495     1
%C A307495     1    1
%C A307495     2    3    3
%C A307495     6    9   12   12
%C A307495    24   33   45   57   57
%C A307495   120  153  198  255  312  312
%C A307495   ...
%C A307495 Alternatively, the sequence can be obtained by multiplying the sequence of factorial numbers by the array A106566.
%C A307495 (End)
%H A307495 P. Bala, <a href="/A001517/a001517.pdf">A note on the Catalan transform of a sequence</a>
%F A307495 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 A307495 Sum_{n>=0} a(n)*(x*(1 - x))^n = g.f. of A000142.
%F A307495 a(n) = (1/n) * Sum_{k=1..n} binomial(2*n-k-1,n-k)*k*k! for n > 0.
%F A307495 a(n) ~ exp(1) * n!. - _Vaclav Kotesovec_, Aug 10 2019
%t A307495 nmax = 23; CoefficientList[Series[Sum[k! ((1 - Sqrt[1 - 4 x])/2)^k, {k, 0, nmax}], {x, 0, nmax}], x]
%t A307495 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 A307495 Join[{1}, Table[1/n Sum[Binomial[2n - k - 1, n - k] k k!, {k, n}], {n, 23}]]
%Y A307495 Cf. A000108, A000142, A013999, A100100, A106566, A307496.
%K A307495 nonn
%O A307495 0,3
%A A307495 _Ilya Gutkovskiy_, Apr 10 2019