A320905 T(n, k) = binomial(2*n - 1 - k, k - 1)*hypergeom([2, 2, 1-k], [1, 1 - 2*k + 2*n], -1), triangle read by rows, T(n, k) for n >= 1 and 1 <= k <= n.

1, 1, 5, 1, 7, 18, 1, 9, 31, 56, 1, 11, 48, 111, 160, 1, 13, 69, 198, 351, 432, 1, 15, 94, 325, 699, 1023, 1120, 1, 17, 123, 500, 1280, 2223, 2815, 2816, 1, 19, 156, 731, 2186, 4458, 6562, 7423, 6912, 1, 21, 193, 1026, 3525, 8330, 14198, 18324, 18943, 16640

Offset: 1

Author

Peter Luschny, Oct 28 2018

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[1] 1
[2] 1,  5
[3] 1,  7,  18
[4] 1,  9,  31,  56
[5] 1, 11,  48, 111,  160
[6] 1, 13,  69, 198,  351,  432
[7] 1, 15,  94, 325,  699, 1023, 1120
[8] 1, 17, 123, 500, 1280, 2223, 2815, 2816
[9] 1, 19, 156, 731, 2186, 4458, 6562, 7423, 6912
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Crossrefs

Row sums with shifted indices in A318947.

T(n, n) = A001793(n).

Programs

Maple

T := (n, k) -> binomial(2*n-1-k,k-1)*hypergeom([2,2,1-k], [1,1-2*k+2*n], -1):
seq(seq(simplify(T(n, k)), k=1..n), n=1..10);

Mathematica

T[n_, k_] := Sum[Binomial[2*n-k, 2*n-2*k+1+j]*Binomial[j+2, 2],{j, 0, 2*n-k}]; Flatten[Table[T[n, k], {n, 1, 10}, {k, 1, n}]] (* Detlef Meya, Dec 31 2023 *)

PARI

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

Python

from functools import cache
@cache
def T(n, k):
    if k < 1 or n < 1: return 0
    if k == 1: return 1
    if k == n: return n * (n + 3) * 2**(n - 3)
    return T(n-1, k) + 2*T(n-1, k-1) - T(n-2, k-2)
for n in range(1, 10): print([T(n, k) for k in range(1, n+1)])
# after Detlef Meya, Peter Luschny, Jan 01 2024

Formula

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