A371026 Triangle read by rows: T(n, k) = 4^n*Sum_{j=0..k} (-1)^(k - j)*binomial(k, j)* Pochhammer(j/4, n).

1, 0, 1, 0, 5, 2, 0, 45, 30, 6, 0, 585, 510, 180, 24, 0, 9945, 10350, 4950, 1200, 120, 0, 208845, 247590, 144900, 48600, 9000, 720, 0, 5221125, 6855030, 4655070, 1940400, 504000, 75600, 5040, 0, 151412625, 216093150, 164872260, 80713080, 26334000, 5594400, 705600, 40320

Offset: 0

Author

Peter Luschny, Mar 08 2024

Keywords

• nonn
• tabl

Examples

			Triangle read by rows:
  [0] 1;
  [1] 0,       1;
  [2] 0,       5,       2;
  [3] 0,      45,      30,       6;
  [4] 0,     585,     510,     180,      24;
  [5] 0,    9945,   10350,    4950,    1200,    120;
  [6] 0,  208845,  247590,  144900,   48600,   9000,   720;
  [7] 0, 5221125, 6855030, 4655070, 1940400, 504000, 75600, 5040;
		

Crossrefs

Cf. A000142 (main diagonal), A007696 (column 1), A371027 (row sums).

Cf. A371025.

Programs

Maple

A371026 := (n, k) -> local j; 4^n*add((-1)^(k - j)*binomial(k, j)*pochhammer(j/4, n), j = 0..k): seq(seq(A371026(n, k), k = 0..n), n = 0..9);

Python

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) + (4*n - 4 + 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) + (4*n - 4 + 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