A297898 Triangle read by rows, T(n, k) = (-1)^(n-k)*binomial(n,k)*hypergeom([k - n, n + 1], k + 1, 2), for n >= 0 and 0 <= k <= n.

1, 3, 1, 13, 4, 1, 63, 19, 5, 1, 321, 96, 26, 6, 1, 1683, 501, 138, 34, 7, 1, 8989, 2668, 743, 190, 43, 8, 1, 48639, 14407, 4043, 1059, 253, 53, 9, 1, 265729, 78592, 22180, 5908, 1462, 328, 64, 10, 1, 1462563, 432073, 122468, 33028, 8378, 1966, 416, 76, 11, 1

Offset: 0

Author

Peter Luschny, Jan 08 2018

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[0]    1
[1]    3,    1
[2]   13,    4,   1
[3]   63,   19,   5,   1
[4]  321,   96,  26,   6,  1
[5] 1683,  501, 138,  34,  7, 1
[6] 8989, 2668, 743, 190, 43, 8, 1
		

Crossrefs

T(n, 0) = A001850(n).

Row sums are A050146(n+1).

Programs

Mathematica

T[n_, k_] := (-1)^(n - k) Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 1, 2];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
T[n_, k_] := Sum[Binomial[n - k, j]*Binomial[n + j, j], {j, 0, n - k}]; Flatten[Table[T[n, k], {n, 0, 9}, {k, 0, n}]] (* Detlef Meya, Jan 14 2024 *)

Formula

T(n, k) = Sum_{j=0..n - k} binomial(n - k, j)*binomial(n + j, j). - Detlef Meya, Jan 14 2024