A190725 Diagonal sums of Riordan matrix A118384.
1, 3, 14, 69, 355, 1872, 10037, 54459, 298138, 1643565, 9111191, 50739120, 283635481, 1590648819, 8945090870, 50423423685, 284831065723, 1611918320688, 9137141645645, 51869777201595, 294843392318146, 1677980087882013, 9559901907126959
Offset: 0
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Programs
-
Mathematica
CoefficientList[Series[(3+3x-Sqrt[1-6x+x^2])/(2(1+3x+x^2)Sqrt[1-6x+x^2]),{x,0,100}],x] -
PARI
x='x+O('x^50); Vec((3+3*x-sqrt(1-6*x+x^2))/(2*(1+3*x+x^2)*sqrt(1-6*x+x^2))) \\ G. C. Greubel, Mar 26 2017
Formula
a(n) = (3*sum((-1)^k*F(2k+1)*d(n-k),k=0..n)-(-1)^n*F(2n+2))/2, where d(n) = central Delannoy number (A001850) and F(n) = Fibonacci number (A000045).
G.f.: (3+3*x-sqrt(1-6*x+x^2))/(2*(1+3*x+x^2)*sqrt(1-6*x+x^2)).
Recurrence: (n^2+11*n+30)*a(n+6)-(3*n^2+29*n+70)*a(n+5)-(17*n^2+177*n+458)*a(n+4)-34*(n+4)*a(n+3)+(17*n^2+95*n+130)*a(n+2)+(3*n^2+19*n+30)*a(n+1)-(n^2+5*n+6)*a(n)=0.
conjecture: n*(2*n-3)*a(n) +(-6*n^2+15*n-8)*a(n-1) +2*(-16*n^2+32*n-11)*a(n-2) +(-6*n^2+9*n-2)*a(n-3) +(2*n-1)*(n-2)*a(n-4) =0. - R. J. Mathar, Jul 24 2012
a(n) ~ sqrt(24+17*sqrt(2))*(3+2*sqrt(2))^n/(6*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 24 2012