A188513 Riordan matrix (1/(x+sqrt(1-4x)),(1-sqrt(1-4x))/(2(x+sqrt(1-4x)))).
1, 1, 1, 3, 3, 1, 9, 11, 5, 1, 29, 40, 23, 7, 1, 97, 147, 99, 39, 9, 1, 333, 544, 413, 194, 59, 11, 1, 1165, 2025, 1691, 907, 333, 83, 13, 1, 4135, 7575, 6842, 4078, 1725, 524, 111, 15, 1, 14845, 28455, 27464, 17856, 8453, 2979, 775, 143, 17, 1, 53791, 107277, 109631, 76718, 39851, 15804, 4797, 1094, 179, 19, 1
Offset: 0
Examples
Triangle begins: 1 1, 1 3, 3, 1 9, 11, 5, 1 29, 40, 23, 7, 1 97, 147, 99, 39, 9, 1 333, 544, 413, 194, 59, 11, 1 1165, 2025, 1691, 907, 333, 83, 13, 1 4135, 7575, 6842, 4078, 1725, 524, 111, 15, 1
Programs
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Mathematica
Flatten[Table[Sum[Binomial[i+k,k]Binomial[2n-i,n+k+i](2k+3i+1)/(n+k+i+1),{i,0,Floor[(n-k)/2]}],{n,0,10},{k,0,n}]] -
Maxima
create_list(sum(binomial(i+k,k)*binomial(2*n-i,n+k+i)*(2*k+3*i+1)/(n+k+i+1),i,0,floor((n-k)/2)),n,0,10,k,0,n);
Formula
T(n,k) = [x^n] ((1-sqrt(1-4*x))/(2*(x+sqrt(1-4*x))))^k/(x+sqrt(1-4*x)).
T(n,k) = [x^(n-k)] (1-2*x)/((1-x)^(n+1)*(1-x-x^2)^(k+1)).
T(n,k) = sum(binomial(i+k,k)*binomial(2*n-i,n+k+i)*(2*k+3*i+1)/(n+k+i+1), i=0..floor((n-k)/2)).
Comments