A375855 Triangle read by rows: T(n, k) = 2^k * hypergeom([-n, -k], [], -1/2).

1, 1, 1, 1, 0, -2, 1, -1, -2, 2, 1, -2, 0, 8, 8, 1, -3, 4, 8, -24, -88, 1, -4, 10, -4, -56, 32, 592, 1, -5, 18, -34, -40, 312, 400, -3344, 1, -6, 28, -88, 96, 512, -1472, -6144, 14464, 1, -7, 40, -172, 448, 32, -4544, 4160, 63616, -2944

Offset: 0

Author

Detlef Meya, Aug 31 2024

Keywords

• sign
• tabl
• easy

Examples

			Triangle starts:
[0] 1;
[1] 1,  1;
[2] 1,  0, -2;
[3] 1, -1, -2,    2;
[4] 1, -2,  0,    8,   8;
[5] 1, -3,  4,    8, -24, -88;
[6] 1, -4, 10,   -4, -56,  32,   592;
[7] 1, -5, 18,  -34, -40, 312,   400, -3344;
[8] 1, -6, 28,  -88,  96, 512, -1472, -6144, 14464;
[9] 1, -7, 40, -172, 448,  32, -4544,  4160, 63616, -2944;
...
		

Links

• John Tyler Rascoe, Rows n = 0..140, flattened

Crossrefs

Cf. A375854, A000012, A295382 (main diagonal).

Programs

Maple

T := (n, k) -> 2^k * hypergeom([-n, -k], [], -1/2);
for n from 0 to 9 do seq(simplify(T(n, k)), k=0..n) od; # Peter Luschny, Sep 02 2024

Mathematica

T[n_, k_] := (-1)^k*Sum[(-2)^(k - j)*Binomial[n, j]*Binomial[k, j]*j!, {j, 0, k}];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten

Python

from math import comb, factorial
def A375855(n,k):
    return (-1)**k*sum((-2)**(k-j)*comb(n, j)*comb(k, j)*factorial(j) for j in range(k+1)) # John Tyler Rascoe, Sep 05 2024

Formula

T(n, k) = (-1)^k*Sum_{j=0..k} (-2)^(k - j)*binomial(n, j)*binomial(k, j)*j!.