A371076 Triangle read by rows: T(n, k) = 3^n*Sum_{j=0..k} (-1)^(k - j)*binomial(k, j) * Pochhammer(j/3, n).

1, 0, 1, 0, 4, 2, 0, 28, 24, 6, 0, 280, 320, 144, 24, 0, 3640, 5040, 3120, 960, 120, 0, 58240, 92960, 71280, 30720, 7200, 720, 0, 1106560, 1975680, 1775760, 960960, 319200, 60480, 5040, 0, 24344320, 47653760, 48545280, 31127040, 13104000, 3548160, 564480, 40320

Offset: 0

Author

Peter Luschny, Mar 10 2024

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[0] 1;
[1] 0,       1;
[2] 0,       4,       2;
[3] 0,      28,      24,       6;
[4] 0,     280,     320,     144,     24;
[5] 0,    3640,    5040,    3120,    960,    120;
[6] 0,   58240,   92960,   71280,  30720,   7200,   720;
[7] 0, 1106560, 1975680, 1775760, 960960, 319200, 60480, 5040;
		

Crossrefs

Cf. A371077, A007559 (column 1), A000142 (main diagonal), A052609 (subdiagonal).

Programs

Maple

A371076 := (n, k) -> local j; 3^n*add((-1)^(k - j)*binomial(k, j)*pochhammer(j/3, n), j = 0..k): seq(seq(A371076(n, k), k = 0..n), n = 0..9);

Formula

T(n, k) = k * T(n-1, k-1) + (3*n - 3 + k) * T(n-1, k) for 0 < k < n with initial values T(n, 0) = 0 for n > 0 and T(n, n) = n! for n >= 0. - Werner Schulte, Mar 13 2024