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%I #41 Jan 30 2020 21:29:15
%S 1,3,11,53,345,2947,31411,400437,5927921,99816515,1882741659,
%T 39310397557,899919305929,22410922177347,603120939234755,
%U 17441737474345973,539390080299331809,17762381612118471043
%N E.g.f.: exp(2*x)/sqrt(1-2*x).
%H Vincenzo Librandi, <a href="/A081367/b081367.txt">Table of n, a(n) for n = 0..99</a>
%F a(n) = Sum_{k = 0..n} A046716(n, k)*2^k. - _Philippe Deléham_, Jun 12 2004
%F a(n) = U(1/2,3/2+n,1)*2^n, where U is the confluent hypergeometric Kummer function U. - _John M. Campbell_, May 04 2011
%F D-finite with recurrence: a(n) = (2*n+1)*a(n-1) - 4*(n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 13 2012
%F a(n) ~ 2^(n+1/2)*n^n/exp(n-1). - _Vaclav Kotesovec_, Oct 13 2012
%F G.f.: W(0)/(1-2*x), where W(k) = 1 - x*(k+1)/(x*(k+1) - (1-2*x)/W(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 03 2014
%F From _Robert Israel_, Nov 04 2014: (Start)
%F a(n) = 2^n * hypergeom([1/2,-n],[],-1).
%F G.f. satisfies (1-3*x+4*x^2)*g(x) + (-2*x^2+4*x^3)*g'(x) = 1. (End)
%p F:= gfun:-rectoproc({a(n) = (2*n+1)*a(n-1) - 4*(n-1)*a(n-2), a(0)=1,a(1)=3},a(n),remember):
%p seq(F(n),n=0..30); # _Robert Israel_, Nov 04 2014
%t Table[HypergeometricU[1/2, 3/2 + n, 1]*2^n, {n, 0, 20}]
%t With[{nn=20},CoefficientList[Series[Exp[2x]/Sqrt[1-2x],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Mar 20 2015 *)
%o (PARI) a(n)=n!*polcoeff(exp(2*x)/sqrt(1-2*x)+O(x^(n+1)),n)
%K nonn
%O 0,2
%A _Benoit Cloitre_, May 10 2003
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