A119307 Triangle read by rows: T(n, k) = Sum_{j=0..n} C(j, k)*C(j, n - k).

1, 1, 1, 1, 5, 1, 1, 11, 11, 1, 1, 19, 46, 19, 1, 1, 29, 127, 127, 29, 1, 1, 41, 281, 517, 281, 41, 1, 1, 55, 541, 1579, 1579, 541, 55, 1, 1, 71, 946, 4001, 6376, 4001, 946, 71, 1, 1, 89, 1541, 8889, 20626, 20626, 8889, 1541, 89, 1, 1, 109, 2377, 17907, 56904, 82994

Offset: 0

Author

Paul Barry, May 13 2006

Keywords

• easy
• nonn
• tabl

Examples

			Triangle begins
  1,
  1, 1,
  1, 5, 1,
  1, 11, 11, 1,
  1, 19, 46, 19, 1,
  1, 29, 127, 127, 29, 1,
  1, 41, 281, 517, 281, 41, 1
  ...
		

Links

• Indranil Ghosh, Rows 0..100, flattened

Crossrefs

Second column is A028387.

Row sums are A014300.

Central coefficients T(2*n, n) are A112029.

Programs

Maple

T := (n, k) -> if n = k then 1 else binomial(n, k)^2*hypergeom([1, -k, -n + k], [-n, -n], 1) fi: for n from 0 to 9 do seq(simplify(T(n, k)), k = 0..n) od;
# Peter Luschny, May 13 2024

Mathematica

Flatten[Table[Sum[Binomial[j,k] Binomial[j,n-k],{j,0,n}],{n,0,10},{k,0,n}]] (* Indranil Ghosh, Mar 03 2017 *)

PARI

tabl(nn)={for (n=0, nn, for(k=0, n, print1(sum(j=0, n, binomial(j,k)*binomial(j,n-k)),", ");); print(););};
tabl(10); \\ Indranil Ghosh, Mar 03 2017

Formula

T(n, k) = T(n, n - k).

T(n, k) = binomial(n, k)^2*hypergeom([1, -k, -n + k], [-n, -n], 1) for k=0..n-1. - Peter Luschny, May 13 2024