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A189175 Riordan matrix (1+x/sqrt(1-4*x),(1-sqrt(1-4*x))/2).

%I A189175 #7 Mar 30 2012 18:55:30
%S A189175 1,1,1,2,2,1,6,5,3,1,20,15,9,4,1,70,49,29,14,5,1,252,168,98,49,20,6,1,
%T A189175 924,594,342,174,76,27,7,1,3432,2145,1221,627,285,111,35,8,1,12870,
%U A189175 7865,4433,2288,1067,440,155,44,9,1,48620,29172,16302,8437,4004,1716,649,209,54,10,1
%N A189175 Riordan matrix (1+x/sqrt(1-4*x),(1-sqrt(1-4*x))/2).
%C A189175 Row sums are A189176, diagonal sums are A189177.
%F A189175 T(n,k)=[x^n] (1+x/sqrt(1-4*x))*((1-sqrt(1-4*x))/2)^k.
%F A189175 T(n,k) = binomial(2*n-k,n-k)*(n^2+n*k-k^2-k)/((2*n-k)*(2*n-k-1)) for k<=n, (n,k) <> (0,0), (1,1).
%F A189175 Recurrence: T(n+1,k+1) = T(n,k) + T(n,k+1) + ... + T(n,n).
%e A189175 Triangle begins:
%e A189175 1
%e A189175 1,1
%e A189175 2,2,1
%e A189175 6,5,3,1
%e A189175 20,15,9,4,1
%e A189175 70,49,29,14,5,1
%e A189175 252,168,98,49,20,6,1
%e A189175 924,594,342,174,76,27,7,1
%e A189175 3432,2145,1221,627,285,111,35,8,1
%t A189175 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 A189175 Flatten[Table[T[n,k],{n,0,20},{k,0,n}]]
%o A189175 (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 A189175 create_list(T(n,k),n,0,20,k,0,n);
%Y A189175 Cf. A189176, A189177.
%K A189175 nonn,easy,tabl
%O A189175 0,4
%A A189175 _Emanuele Munarini_, Apr 18 2011