A375612 Triangle read by rows: T(n, k) = n! * 4^k * hypergeom([-k], [-n], -1/4).

1, 1, 3, 2, 7, 25, 6, 22, 81, 299, 24, 90, 338, 1271, 4785, 120, 456, 1734, 6598, 25121, 95699, 720, 2760, 10584, 40602, 155810, 598119, 2296777, 5040, 19440, 75000, 289416, 1117062, 4312438, 16651633, 64309755, 40320, 156240, 605520, 2347080, 9098904, 35278554, 136801778, 530555479, 2057912161

Offset: 0

Author

Detlef Meya, Aug 21 2024

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[0] 1;
[1] 1, 3;
[2] 2, 7, 25;
[3] 6, 22, 81, 299;
[4] 24, 90, 338, 1271, 4785;
[5] 120, 456, 1734, 6598, 25121, 95699;
[6] 720, 2760, 10584, 40602, 155810, 598119, 2296777;
[7] 5040, 19440, 75000, 289416, 1117062, 4312438, 16651633, 64309755;
...
		

Crossrefs

Cf. A375613, A000142, A001907 (main diagonal).

Cf. A374427, A374428, A375446, A375447, A375597, A375600.

Programs

Mathematica

T[n_, k_] := (-1)^k*Sum[(-4)^(k - j)*Binomial[k, k - j]*(n - j)!, {j, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

Formula

T(n, k) = (-1)^k*Sum_{j=0..k} (-4)^(k - j)*binomial(k, k - j)*(n - j)!.