A320906 T(n, k) = binomial(2*n - k, k - 1)*hypergeom([2, 2, 1 - k], [1, 2*(1 - k + n)], -1), triangle read by rows, T(n,k) for n >= 0 and 0 <= k <= n.

0, 0, 1, 0, 1, 6, 0, 1, 8, 24, 0, 1, 10, 39, 80, 0, 1, 12, 58, 150, 240, 0, 1, 14, 81, 256, 501, 672, 0, 1, 16, 108, 406, 955, 1524, 1792, 0, 1, 18, 139, 608, 1686, 3178, 4339, 4608, 0, 1, 20, 174, 870, 2794, 6144, 9740, 11762, 11520

Offset: 0

Author

Peter Luschny, Oct 28 2018

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[0] 0
[1] 0, 1
[2] 0, 1,  6
[3] 0, 1,  8,  24
[4] 0, 1, 10,  39,  80
[5] 0, 1, 12,  58, 150,  240
[6] 0, 1, 14,  81, 256,  501,  672
[7] 0, 1, 16, 108, 406,  955, 1524, 1792
[8] 0, 1, 18, 139, 608, 1686, 3178, 4339,  4608
[9] 0, 1, 20, 174, 870, 2794, 6144, 9740, 11762, 11520
		

Links

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

Crossrefs

Row sums are A320907. T(n, n) = A001788(n).

Cf. A320905.

Programs

Maple

T := (n, k) -> binomial(2*n-k, k-1)*hypergeom([2, 2, 1-k], [1, 2*(1-k+n)], -1):
seq(seq(simplify(T(n, k)), k=0..n), n=0..9);

Mathematica

T[n_, k_] := Sum[Binomial[2*n+1-k, 2*n+2-2*k+j]*Binomial[j+2, 2], {j,0, 2*n+1-k}]; Flatten[Table[T[n, k], {n, 0, 15}, {k, 0, n}]] (* Detlef Meya, Dec 31 2023 *)

PARI

T(n, k) = {sum(j=0, 2*n+1-k, binomial(2*n+1-k, 2*n+2-2*k+j) * binomial(j+2,2))} \\ Andrew Howroyd, Dec 31 2023

Python

from functools import cache
@cache
def T(n, k):
    if k <= 0 or n <= 0: return 0
    if k == 1: return 1
    if k == n: return n * (n + 1) * 2**(n - 2)
    return T(n-1, k) + 2*T(n-1, k-1) - T(n-2, k-2)
for n in range(10): print([T(n, k) for k in range(n + 1)])
# after Detlef Meya, Peter Luschny, Jan 01 2024

Formula

T(n, k) = Sum_{j=0..2*n+1-k} binomial(2*n+1-k, 2*n+2-2*k+j) * binomial(j+2,2). - Detlef Meya, Dec 31 2023