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
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;
Programs
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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);
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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