A190215 Riordan matrix ((1-x-x^2)/(1-2x-x^2),(x-x^2-x^3)/(1-2x-x^2)).
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, 169, 271, 246, 157, 75, 27, 7, 1, 408, 714, 707, 496, 265, 110, 35, 8, 1, 985, 1868, 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
Offset: 0
Examples
Triangle begins:
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;
169, 271, 246, 157, 75, 27, 7, 1;
408, 714, 707, 496, 265, 110, 35, 8, 1;
Links
- G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Programs
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Mathematica
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}]] -
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);
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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
Formula
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) )).
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).
Comments