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%I #16 Jul 22 2015 15:47:57
%S 1,2,5,14,42,132,430,1444,4984,17648,64024,237712,902416,3499680,
%T 13853424,55931168,230142848,964460288,4113656704,17846729984,
%U 78708574976,352678567424,1604739694848,7411167960576,34723660917760
%N Row sums of factorial-Pascal matrix A162747.
%C Second binomial transform of aerated factorial numbers. Binomial transform of A084261. Hankel transform is A137704.
%H Vincenzo Librandi, <a href="/A162748/b162748.txt">Table of n, a(n) for n = 0..200</a>
%F G.f.: 1/(1-2x-x^2/(1-2x-x^2/(1-2x-2x^2/(1-2x-2x^2/(1-2x-3x^2/(1-2x-3x^2/(1-2x-4x^2/(1-2x-... (continued fraction);
%F a(n)=sum{k=0..floor(n/2), C(n,2k)*2^(n-2k)*F(k+1)}=sum{k=0..n, C(n,k)*2^(n-k)*(k/2)!*(1+(-1)^k)/2}.
%F a(n)=sum{k=0..n, A161556(n,k)*2^k}. - _Paul Barry_, Apr 11 2010
%F E.g.f.: exp(2x)*(1+(sqrt(Pi)/2)*x*exp(x^2/4)*erf(x/2)). - _Paul Barry_, Sep 17 2010
%F Apparently -2*a(n) +8*a(n-1) +(n-8)*a(n-2) +2*(2-n)*a(n-3)=0. - _R. J. Mathar_, Oct 25 2012
%F a(n) ~ 1/2 * sqrt(Pi*n) * exp(2*sqrt(2*n)-n/2-2) * (n/2)^(n/2) * (1 + 1/(3*sqrt(2*n))). - _Vaclav Kotesovec_, Aug 15 2013
%t Table[Sum[Binomial[n,k]*2^(n-k)*(k/2)!*(1+(-1)^k)/2,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Aug 15 2013 *)
%K easy,nonn
%O 0,2
%A _Paul Barry_, Jul 12 2009
%E Minor edits by _Vaclav Kotesovec_, Jul 22 2015
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