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

%I A189177 #12 Oct 20 2024 03:34:26
%S A189177 1,1,3,8,26,88,311,1125,4139,15411,57901,219070,833509,3185834,
%T A189177 12223298,47048989,181596815,702589992,2723964698,10580344863,
%U A189177 41163089721,160380285133,625698670720,2443971874519,9556455398766,37404704660996,146537441530003,574556162261917
%N A189177 Diagonal sums of the Riordan matrix (1+x/sqrt(1-4*x),(1-sqrt(1-4*x))/2) (A189175).
%H A189177 Vincenzo Librandi, <a href="/A189177/b189177.txt">Table of n, a(n) for n = 0..100</a>
%F A189177 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.
%F A189177 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)).
%t A189177 T[n_,k_]=If[n==k,1,Binomial[2n-k,n-k](n^2+n k-k^2-k)/((2n-k)(2n-k-1))]
%t A189177 Table[Sum[T[n-k,k],{k,0,Floor[n/2]}],{n,0,22}]
%o A189177 (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));
%o A189177 makelist(sum(T(n-k,k),k,0,floor(n/2)),n,0,22);
%Y A189177 Cf. A189175, A189176.
%K A189177 nonn,easy
%O A189177 0,3
%A A189177 _Emanuele Munarini_, Apr 18 2011