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%I #34 Jul 27 2022 08:25:20
%S 1,1,5,25,121,641,4861,44185,272945,480961,19687861,754778201,
%T 1734823465,-259904463935,180602875181,188945735209561,21689448762721,
%U -167679491870531455,2951642934531685,193773398162963638681,448885053392410841
%N E.g.f. exp(x*(sqrt(4*x^2+1)+2*x)).
%F a(n) = (n-1)!*sum(k = 1..n, (4^(n-k)*binomial(n/2,n-k))/(k-1)!),n>0, a(0)=1.
%F a(n) ~ 2^((3*n-1)/2) * n^(n/2) / exp(n/2-1/4) if n is even, and a(n) ~ (-1)^((n+1)/2) * n^(n-1) * 2^n / exp(n+1/2) if n is odd. - _Vaclav Kotesovec_, Oct 31 2014
%F a(2*n+1) = 16^n*hypergeom([-2*n],[3/2-n],-1/4)*Gamma(3/2+n)/Gamma(3/2-n); a(2*n+1) ~ (-1)^(n+1)*2*(4*n/exp(1))^(2*n)/exp(1/2). - _Peter Luschny_, Oct 31 2014
%F D-finite with recurrence (-2*n+3)*a(n) +(-8*n^3+60*n^2-78*n-15)*a(n-2) +32*(n-2)*(n-3)*(n-4)*(2*n+1)*a(n-4)=0. - _R. J. Mathar_, Jul 27 2022
%p A249454 := proc(n)
%p if n = 0 then
%p 1;
%p else
%p (n-1)!*add( 4^(n-k)*binomial(n/2,n-k)/(k-1)! ,k=1..n) ;
%p end if;
%p end proc;
%p seq(A249454(n),n=0..40) ; # _R. J. Mathar_, Jul 27 2022
%t CoefficientList[Series[E^(x*(Sqrt[4*x^2+1]+2*x)), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Oct 31 2014 *)
%o (Maxima)
%o a(n):=if n=0 then 1 else (n-1)!*sum((4^(n-k)*binomial(n/2,n-k))/(k-1)!,k,1,n);
%K sign
%O 0,3
%A _Vladimir Kruchinin_, Oct 31 2014
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