A371025 Triangle read by rows: T(n, k) = 2^n*Sum_{j=0..k} (-1)^(k - j)*binomial(k, j)* Pochhammer(j/2, n).

1, 0, 1, 0, 3, 2, 0, 15, 18, 6, 0, 105, 174, 108, 24, 0, 945, 1950, 1710, 720, 120, 0, 10395, 25290, 28080, 16920, 5400, 720, 0, 135135, 374850, 497070, 383040, 176400, 45360, 5040, 0, 2027025, 6267870, 9574740, 8883000, 5266800, 1965600, 423360, 40320

Offset: 0

Author

Peter Luschny, Mar 08 2024

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[0] 1;
[1] 0,      1;
[2] 0,      3,      2;
[3] 0,     15,     18,      6;
[4] 0,    105,    174,    108,     24;
[5] 0,    945,   1950,   1710,    720,    120;
[6] 0,  10395,  25290,  28080,  16920,   5400,   720;
[7] 0, 135135, 374850, 497070, 383040, 176400, 45360, 5040;
		

Crossrefs

Cf. A000142 (main diagonal), A001147 (column 1), A308939 (row sums).

Programs

Maple

A371025 := (n, k) -> local j; 2^n*add((-1)^(k - j)*binomial(k, j)*pochhammer(j/2, n), j = 0..k); seq(seq(A371025(n, k), k = 0..n), n = 0..9);

SageMath

from functools import cache
@cache
def T(n, k):   # after Werner Schulte
    if k == 0: return 0**n
    if k == n: return n * T(n-1, n-1)
    return k * T(n-1, k-1) + (2*n - 2 + k) * T(n-1, k)
for n in range(8): print([T(n, k) for k in range(n + 1)])
# Peter Luschny, Mar 17 2024

Formula

T(n, k) = k * T(n-1, k-1) + (2*n - 2 + 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 17 2024