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

0,5

Links

Indranil Ghosh, Rows 0..100, flattened

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

Example

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
  ...

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 *)

Prog

(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

Crossrefs

Second column is A028387.

Row sums are A014300.

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

Sequence in context: A082046 A132787 A181370 * A296039 A296974 A146954

Adjacent sequences: A119304 A119305 A119306 * A119308 A119309 A119310

Keyword

easy,nonn,tabl

Author

Paul Barry, May 13 2006

Status

approved