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

%I A371026 #10 Mar 17 2024 18:32:05
%S A371026 1,0,1,0,5,2,0,45,30,6,0,585,510,180,24,0,9945,10350,4950,1200,120,0,
%T A371026 208845,247590,144900,48600,9000,720,0,5221125,6855030,4655070,
%U A371026 1940400,504000,75600,5040,0,151412625,216093150,164872260,80713080,26334000,5594400,705600,40320
%N A371026 Triangle read by rows: T(n, k) = 4^n*Sum_{j=0..k} (-1)^(k - j)*binomial(k, j)* Pochhammer(j/4, n).
%F A371026 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
%e A371026 Triangle read by rows:
%e A371026   [0] 1;
%e A371026   [1] 0,       1;
%e A371026   [2] 0,       5,       2;
%e A371026   [3] 0,      45,      30,       6;
%e A371026   [4] 0,     585,     510,     180,      24;
%e A371026   [5] 0,    9945,   10350,    4950,    1200,    120;
%e A371026   [6] 0,  208845,  247590,  144900,   48600,   9000,   720;
%e A371026   [7] 0, 5221125, 6855030, 4655070, 1940400, 504000, 75600, 5040;
%p A371026 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);
%o A371026 (Python)
%o A371026 from functools import cache
%o A371026 @cache
%o A371026 def T(n, k):   # After _Werner Schulte_
%o A371026     if k == 0: return 0**n
%o A371026     if k == n: return n * T(n-1, n-1)
%o A371026     return k * T(n-1, k-1) + (4*n - 4 + k) * T(n-1, k)
%o A371026 for n in range(8): print([T(n, k) for k in range(n + 1)])
%o A371026 # _Peter Luschny_, Mar 17 2024
%Y A371026 Cf. A000142 (main diagonal), A007696 (column 1), A371027 (row sums).
%Y A371026 Cf. A371025.
%K A371026 nonn,tabl
%O A371026 0,5
%A A371026 _Peter Luschny_, Mar 08 2024