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A293468 a(n) = Sum_{k=0..n} k!*binomial(2*n-k, n).

%I A293468 #15 Oct 18 2017 05:05:08
%S A293468 1,3,11,44,189,880,4542,26712,182793,1461368,13477650,140564536,
%T A293468 1627370146,20621925504,283161372284,4182215376240,66065933347425,
%U A293468 1111053154779720,19814069772086730,373435157945506680,7415765258637418950,154751460071567005920,3385387828167428482020
%N A293468 a(n) = Sum_{k=0..n} k!*binomial(2*n-k, n).
%H A293468 Robert Israel, <a href="/A293468/b293468.txt">Table of n, a(n) for n = 0..449</a>
%H A293468 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A293468 a(n) = [x^n] 1/((1 - x)^(n+1)*(1 - x/(1 - x/(1 - 2*x/(1 - 2*x/(1 - 3*x/(1 - 3*x/(1 - ...)))))))), a continued fraction.
%F A293468 a(n) = Gamma(n+1/2)*4^n*hypergeom([1,1,-n],[-2n],1)/(sqrt(Pi)*n!). - _Robert Israel_, Oct 09 2017
%F A293468 a(n) ~ exp(1) * n!. - _Vaclav Kotesovec_, Oct 18 2017
%p A293468 seq(simplify( GAMMA(n+1/2)*4^n*hypergeom([1,1,-n],[-2*n],1)/(sqrt(Pi)*n!)),n=0..30); # _Robert Israel_, Oct 09 2017
%t A293468 Table[Sum[k! Binomial[2 n - k, n], {k, 0, n}], {n, 0, 22}]
%t A293468 Table[Sum[Gamma[k + 1] Gamma[2 n - k  + 1]/(Gamma[n + 1] Gamma[n - k + 1]), {k, 0, n}], {n, 0, 22}]
%t A293468 Table[SeriesCoefficient[(1/(1 - x)^(n + 1)) 1/(1 + ContinuedFractionK[-Floor[(k + 1)/2] x, 1, {k, 1, n}]), {x, 0, n}], {n, 0, 22}]
%t A293468 Table[SeriesCoefficient[(1/(1 - x)^(n + 1)) Sum[k! x^k, {k, 0, n}], {x, 0, n}], {n, 0, 22}]
%t A293468 A293468[n_] := DifferenceRoot[Function[{a,k}, {(k+1)(k-n)a[k] + (k(n-2)-k^2+3n)
%t A293468 a[k+1] + (k-2n) a[k+2] == 0, a[0] == 0, a[1] == Binomial[2n, n]}]][1+n];
%t A293468 Table[A293468[n], {n, 0, 22}] (* _Peter Luschny_, Oct 09 2017 *)
%Y A293468 Cf. A000142, A000522, A003422, A014144, A092392, A270447, A293469.
%K A293468 nonn
%O A293468 0,2
%A A293468 _Ilya Gutkovskiy_, Oct 09 2017