A189177 Diagonal sums of the Riordan matrix (1+x/sqrt(1-4*x),(1-sqrt(1-4*x))/2) (A189175).

1, 1, 3, 8, 26, 88, 311, 1125, 4139, 15411, 57901, 219070, 833509, 3185834, 12223298, 47048989, 181596815, 702589992, 2723964698, 10580344863, 41163089721, 160380285133, 625698670720, 2443971874519, 9556455398766, 37404704660996, 146537441530003, 574556162261917

Offset: 0

Author

Emanuele Munarini, Apr 18 2011

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A189175, A189176.

Programs

Mathematica

T[n_,k_]=If[n==k,1,Binomial[2n-k,n-k](n^2+n k-k^2-k)/((2n-k)(2n-k-1))]
Table[Sum[T[n-k,k],{k,0,Floor[n/2]}],{n,0,22}]

Maxima

T(n,k):=if n=k then 1 else binomial(2*n-k,n-k)*(n^2+n*k-k^2-k)/((2*n-k)*(2*n-k-1));
makelist(sum(T(n-k,k),k,0,floor(n/2)),n,0,22);

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-3*k,n-k)*(n^2-n*k-k^2-k)/((2*n-3*k)*(2*n-3*k-1)), for n>=3.

G.f.: (2-9*x+3*x^2+4*x^3+(x+3*x^2)*sqrt(1-4*x))/(2*(1-4*x)*(1-x+x^3)).