A162748 Row sums of factorial-Pascal matrix A162747.
1, 2, 5, 14, 42, 132, 430, 1444, 4984, 17648, 64024, 237712, 902416, 3499680, 13853424, 55931168, 230142848, 964460288, 4113656704, 17846729984, 78708574976, 352678567424, 1604739694848, 7411167960576, 34723660917760
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Programs
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Mathematica
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 *)
Formula
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);
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}.
a(n)=sum{k=0..n, A161556(n,k)*2^k}. - Paul Barry, Apr 11 2010
E.g.f.: exp(2x)*(1+(sqrt(Pi)/2)*x*exp(x^2/4)*erf(x/2)). - Paul Barry, Sep 17 2010
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
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
Extensions
Minor edits by Vaclav Kotesovec, Jul 22 2015
Comments