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

0,2

Comments

Second binomial transform of aerated factorial numbers. Binomial transform of A084261. Hankel transform is A137704.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

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

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 *)

Crossrefs

Sequence in context: A126567 A125501 A126569 * A374548 A061815 A340361

Adjacent sequences: A162745 A162746 A162747 * A162749 A162750 A162751

Keyword

easy,nonn

Author

Paul Barry, Jul 12 2009

Extensions

Minor edits by Vaclav Kotesovec, Jul 22 2015

Status

approved