A189187 Riordan matrix (1/(1-x-x^2-x^3),(x+x^2)/(1-x-x^2-x^3)).
1, 1, 1, 2, 3, 1, 4, 7, 5, 1, 7, 17, 16, 7, 1, 13, 38, 46, 29, 9, 1, 24, 82, 122, 99, 46, 11, 1, 44, 174, 304, 303, 184, 67, 13, 1, 81, 362, 728, 857, 641, 309, 92, 15, 1, 149, 743, 1690, 2291, 2031, 1212, 482, 121, 17, 1, 274, 1509, 3827, 5869, 6004, 4260, 2108, 711, 154, 19, 1
Offset: 0
Examples
Triangle begins: 1 1,1 2,3,1 4,7,5,1 7,17,16,7,1 13,38,46,29,9,1 24,82,122,99,46,11,1 44,174,304,303,184,67,13,1 81,362,728,857,641,309,92,15,1
Programs
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Mathematica
Flatten[Table[Sum[Binomial[i+k,k]Sum[Binomial[i+k,j]Binomial[n-i-j,i+k],{j,0,n-k-2i}],{i,0,n}],{n,0,20},{k,0,n}]] -
Maxima
create_list(sum(binomial(i+k,k)*sum(binomial(i+k,j)*binomial(n-i-j,i+k),j,0,n-k-2*i),i,0,n),n,0,8,k,0,n);
Formula
T(n,k) = [x^n](x+x^2)^k/(1-x-x^2-x^3)^(k+1).
T(n,k) = sum(binomial(i+k,k)*sum(binomial(i+k,j)*binomial(n-i-j,i+k),j=0..n-k-2*i),i=0..n).
T(n,k) = sum(binomial(k,i)*(-1)^(k-i)*sum(binomial(j+k,k)*trinomial(i+j,n-3*k+2*i-j),j=0..n-k),i=0..k)
Recurrence: T(n+3,k+1) = T(n+2,k+1) + T(n+2,k) + T(n+1,k+1) + T(n+1,k) + T(n,k+1)
Extensions
a(23) and a(40) corrected by Georg Fischer, Feb 20 2021 and Apr 29 2022
Comments