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%I #24 Dec 26 2023 10:02:27
%S 1,1,1,2,2,1,5,5,3,1,12,14,9,4,1,29,38,28,14,5,1,70,102,84,48,20,6,1,
%T 169,271,246,157,75,27,7,1,408,714,707,496,265,110,35,8,1,985,1868,
%U 2001,1526,896,417,154,44,9,1,2378,4858,5592,4596,2930,1500,623,208,54,10,1,5741,12569,15461,13602,9330,5186,2373,894,273,65,11,1
%N Riordan matrix ((1-x-x^2)/(1-2x-x^2),(x-x^2-x^3)/(1-2x-x^2)).
%C Row sums = A052963.
%C Diagonal sums = A052960.
%C Central coefficients = A190315.
%H G. C. Greubel, <a href="/A190215/b190215.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F T(n,k) = Sum_{i=0..n-k} (binomial(i+k,k)*Sum_{j=0..n-k-i} (binomial(i+j-1,j)*binomial(j,n-k-i-j) )).
%F Recurrence: T(n+3,k+1) = 2 T(n+2,k+1) + T(n+2,k) + T(n+1,k+1) - T(n+1,k) - T(n,k).
%e Triangle begins:
%e 1;
%e 1, 1;
%e 2, 2, 1;
%e 5, 5, 3, 1;
%e 12, 14, 9, 4, 1;
%e 29, 38, 28, 14, 5, 1;
%e 70, 102, 84, 48, 20, 6, 1;
%e 169, 271, 246, 157, 75, 27, 7, 1;
%e 408, 714, 707, 496, 265, 110, 35, 8, 1;
%t Flatten[Table[Sum[Binomial[i+k,k]Sum[Binomial[i+j-1,j]Binomial[j,n-k-i-j],{j,0,n-k-i}],{i,0,n-k}],{n,0,12},{k,0,n}]]
%o (Maxima) create_list(sum(binomial(i+k,k)*sum(binomial(i+j-1,j)*binomial(j,n-k-i-j),j,0,n-k-i),i,0,n-k),n,0,12,k,0,n);
%o (PARI) for(n=0,10, for(k=0,n, print1(sum(j=0,n-k, binomial(j+k,k)* sum(r=0,n-k-j, binomial(j+r-1,r)*binomial(r,n-k-j-r))), ", "))) \\ _G. C. Greubel_, Dec 27 2017
%Y Cf. A052963, A052960, A190315.
%K nonn,tabl
%O 0,4
%A _Emanuele Munarini_, May 10 2011
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