A116382 Riordan array (1/sqrt(1-4*x^2), (1-2*x^2*c(x^2))*(x^2*c(x^2))/(x*(1-x-x^2*c(x^2)))) where c(x) is the g.f. of A000108.
1, 0, 1, 2, 1, 1, 0, 3, 2, 1, 6, 4, 5, 3, 1, 0, 10, 10, 8, 4, 1, 20, 15, 21, 19, 12, 5, 1, 0, 35, 42, 42, 32, 17, 6, 1, 70, 56, 84, 92, 77, 50, 23, 7, 1, 0, 126, 168, 192, 180, 131, 74, 30, 8, 1, 252, 210, 330, 405, 400, 326, 210, 105, 38, 9, 1
Offset: 0
Examples
Triangle begins
1;
0, 1;
2, 1, 1;
0, 3, 2, 1;
6, 4, 5, 3, 1;
0, 10, 10, 8, 4, 1;
20, 15, 21, 19, 12, 5, 1;
0, 35, 42, 42, 32, 17, 6, 1;
70, 56, 84, 92, 77, 50, 23, 7, 1;
0, 126, 168, 192, 180, 131, 74, 30, 8, 1;
252, 210, 330, 405, 400, 326, 210, 105, 38, 9, 1;
Links
- G. C. Greubel, Rows n = 0..100 of triangle, flattened
Programs
-
GAP
Flat(List([0..10], n-> List([0..n], k-> Sum([0..n], j-> (-1)^(n-j)*Binomial(n,j)*Sum([0..j], m-> Binomial(j,m-k)*Binomial(m,j-m) ))))); # G. C. Greubel, May 22 2019
-
Magma
T:= func< n,k | (&+[(-1)^(n-j)*Binomial(n,j)*(&+[Binomial(j,m-k)* Binomial(m,j-m): m in [0..j]]): j in [0..n]]) >; [[T(n,k): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 22 2019
-
Mathematica
T[n_, k_] := Sum[(-1)^(n-j)*Binomial[n, j]*Sum[Binomial[j, i-k]* Binomial[i, j-i], {i, 0, j}], {j, 0, n}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 24 2018 *) -
PARI
{T(n,k) = sum(j=0,n, (-1)^(n-j)*binomial(n,j)*sum(m=0,j, binomial(j,m-k)*binomial(m,j-m) ))}; \\ G. C. Greubel, May 22 2019 -
Sage
def T(n, k): return sum((-1)^(n-j)*binomial(n,j)*sum(binomial(j,m-k)*binomial(m,j-m) for m in (0..j)) for j in (0..n)) [[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 22 2019
Formula
Riordan array (1/sqrt(1-4*x^2), sqrt(1-4*x^2)*(1-sqrt(1-4*x^2))/(x-2*x^2 + x*sqrt(1-4*x^2))).
Number triangle T(n,k) = Sum{j=0..n} (-1)^(n-j)* C(n,j)*Sum_{i=0..j} C(j,i-k)*C(i,j-i).
Comments