A297899 Triangle read by rows, T(n, k) = binomial(n, k)*hypergeom([k-n, n+1], [k+2], -4), for n >= 0 and 0 <= k <= n.

1, 5, 1, 45, 10, 1, 505, 115, 15, 1, 6345, 1460, 210, 20, 1, 85405, 19765, 2990, 330, 25, 1, 1204245, 279710, 43635, 5220, 475, 30, 1, 17558705, 4088615, 651165, 81955, 8275, 645, 35, 1, 262577745, 61254760, 9901860, 1290520, 139350, 12280, 840, 40, 1

Offset: 0

Author

Peter Luschny, Jan 08 2018

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[0]       1
[1]       5,      1
[2]      45,     10,     1
[3]     505,    115,    15,    1
[4]    6345,   1460,   210,   20,   1
[5]   85405,  19765,  2990,  330,  25,  1
[6] 1204245, 279710, 43635, 5220, 475, 30, 1
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Crossrefs

T(n, 0) = A133305(n). Row sums are A297705, alternating row sums are A131765.

Cf. A103209.

Programs

Mathematica

T[n_, k_] := Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, -4];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
T[n_, k_] := Sum[4^(j - k)*(k + 1)*Binomial[n + j - k, 2*j - k]*Binomial[2*j - k, j - k]/(j + 1), {j, k, n}];
Flatten[Table[T[n, k], {n, 0, 8}, {k, 0, n}]] (* Detlef Meya, Jan 15 2024 *)

PARI

T(n,k) = sum(j = k, n, 4^(j - k)*(k + 1)*binomial(n + j - k, 2*j - k)* binomial(2*j - k, j - k)/(j + 1)) \\ Andrew Howroyd, Jan 15 2024

Formula

T(n, k) = Sum_{j = k..n} 4^(j - k)*(k + 1)*binomial(n + j - k, 2*j - k)* binomial(2*j - k, j - k)/(j + 1). - Detlef Meya, Jan 15 2024