OFFSET
0,5
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
FORMULA
Riordan array (1, x*c(x)*c(x*c(x))), where c(x) is the g.f. of A000108.
T(n,k) = (k/n)*Sum_{i=k..n} C(2*i-k-1,i-k)*C(2*n-i-1,n-i), T(n,n)=1. - Vladimir Kruchinin, Apr 05 2019
EXAMPLE
Triangle begins
1;
0, 1;
0, 2, 1;
0, 6, 4, 1;
0, 21, 16, 6, 1;
0, 80, 66, 30, 8, 1;
0, 322, 280, 143, 48, 10, 1;
0, 1348, 1216, 672, 260, 70, 12, 1;
0, 5814, 5385, 3150, 1344, 425, 96, 14, 1;
0, 25674, 24244, 14799, 6784, 2400, 646, 126, 16, 1;
0, 115566, 110704, 69828, 33814, 13002, 3960, 931, 160, 18, 1;
MATHEMATICA
T[n_, k_]:= If[k==n, 1, (k/n)*Sum[Binomial[2*j-k-1, j-k]*Binomial[2*n-j- 1, n-j], {j, k, n}]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 05 2019 *)
PROG
(Maxima)
T(n, k):=if k=n then 1 else if n=0 then 0 else (k*sum((binomial(-k+2*i-1, i-k))*(binomial(2*n-i-1, n-i)), i, k, n))/n; /* Vladimir Kruchinin, Apr 05 2019 */
(PARI) {T(n, k) = if(k==n, 1, (k/n)*sum(j=0, n-k, binomial(2*j+k-1, j)* binomial(2*n-k-j-1, n-k-j)))}; \\ G. C. Greubel, Apr 05 2019
(Magma) [[k eq n select 1 else (k/n)*(&+[Binomial(2*j+k-1, j)*Binomial(2*n -k-j-1, n-k-j): j in [0..n-k]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Apr 05 2019
(Sage)
def T(n, k):
if k == n: return 1
return (k*sum(binomial(2*j+k-1, j)* binomial(2*n-k-j-1, n-k-j) for j in (0..n-k)))//n
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 05 2019
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Jan 20 2007
STATUS
approved