A116392 Riordan array (1/sqrt(1-2*x-3*x^2), 1/sqrt(1-2*x-3*x^2) -1).
1, 1, 1, 3, 4, 1, 7, 13, 7, 1, 19, 42, 32, 10, 1, 51, 131, 128, 60, 13, 1, 141, 406, 475, 292, 97, 16, 1, 393, 1247, 1685, 1267, 561, 143, 19, 1, 1107, 3814, 5800, 5112, 2804, 962, 198, 22, 1, 3139, 11623, 19540, 19624, 12748, 5464, 1522, 262, 25, 1, 8953, 35334
Offset: 0
Examples
Triangle begins: 1; 1, 1; 3, 4, 1; 7, 13, 7, 1; 19, 42, 32, 10, 1; 51, 131, 128, 60, 13, 1;
Links
- G. C. Greubel, Rows n = 0..100 of triangle, flattened
Programs
-
Magma
[[(&+[ Binomial(n,m)*(&+[ (&+[ Round((-1)^(k-j)*4^r* Binomial(k,j)*Binomial(j, m-2*r)*Gamma(r+(j+1)/2)/(Factorial(r)*Gamma((j+1)/2))) : r in [0..Floor(n/2)]]) : j in [0..k]]): m in [0..n]]) : k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 23 2019
-
Maple
# The function RiordanSquare is defined in A321620. RiordanSquare(1/sqrt(1 - 2*x - 3*x^2), 10); # Peter Luschny, Feb 15 2020
-
Mathematica
t[n_,k_]:= Sum[(-1)^(k-j)*Binomial[k,j]*Sum[4^r*Binomial[r+(j-1)/2, r]* Binomial[j, n-2*r], {r,0,Floor[n/2]}], {j,0,k}]; Table[Sum[Binomial[n, j]*t[j,k], {j,0,n}] {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, May 23 2019 *) -
PARI
t(n,k) = sum(j=0,k, sum(r=0,floor(n/2), (-1)^(k-j)*4^r* binomial(k,j)*binomial(r+(j-1)/2, r)*binomial(j, n-2*r) )); T(n,k) = sum(j=0,n, binomial(n,j)*t(j,k)); \\ G. C. Greubel, May 23 2019
-
Sage
[[sum(binomial(n,m)*sum( sum( (-1)^(k-j)*4^r* binomial(k,j)* binomial(r+(j-1)/2, r)*binomial(j, m-2*r) for r in (0..floor(n/2))) for j in (0..k)) for m in (0..n)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 23 2019
Formula
Number triangle T(n,k) = Sum_{j=0..n} C(n,j)*A116389(j,k).
Comments