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A190215 Riordan matrix ((1-x-x^2)/(1-2x-x^2),(x-x^2-x^3)/(1-2x-x^2)).

%I A190215 #24 Dec 26 2023 10:02:27
%S A190215 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 A190215 169,271,246,157,75,27,7,1,408,714,707,496,265,110,35,8,1,985,1868,
%U A190215 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 A190215 Riordan matrix ((1-x-x^2)/(1-2x-x^2),(x-x^2-x^3)/(1-2x-x^2)).
%C A190215 Row sums = A052963.
%C A190215 Diagonal sums = A052960.
%C A190215 Central coefficients = A190315.
%H A190215 G. C. Greubel, <a href="/A190215/b190215.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A190215 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 A190215 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 A190215 Triangle begins:
%e A190215     1;
%e A190215     1,   1;
%e A190215     2,   2,   1;
%e A190215     5,   5,   3,   1;
%e A190215    12,  14,   9,   4,   1;
%e A190215    29,  38,  28,  14,   5,   1;
%e A190215    70, 102,  84,  48,  20,   6,   1;
%e A190215   169, 271, 246, 157,  75,  27,   7,   1;
%e A190215   408, 714, 707, 496, 265, 110,  35,   8,   1;
%t A190215 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 A190215 (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 A190215 (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 A190215 Cf. A052963, A052960, A190315.
%K A190215 nonn,tabl
%O A190215 0,4
%A A190215 _Emanuele Munarini_, May 10 2011