A371026 - OEIS

%I #10 Mar 17 2024 18:32:05

%S 1,0,1,0,5,2,0,45,30,6,0,585,510,180,24,0,9945,10350,4950,1200,120,0,

%T 208845,247590,144900,48600,9000,720,0,5221125,6855030,4655070,

%U 1940400,504000,75600,5040,0,151412625,216093150,164872260,80713080,26334000,5594400,705600,40320

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

%F 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 Triangle read by rows:

%e   [0] 1;

%e   [1] 0,       1;

%e   [2] 0,       5,       2;

%e   [3] 0,      45,      30,       6;

%e   [4] 0,     585,     510,     180,      24;

%e   [5] 0,    9945,   10350,    4950,    1200,    120;

%e   [6] 0,  208845,  247590,  144900,   48600,   9000,   720;

%e   [7] 0, 5221125, 6855030, 4655070, 1940400, 504000, 75600, 5040;

%p 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 (Python)

%o from functools import cache

%o @cache

%o def T(n, k):   # After _Werner Schulte_

%o     if k == 0: return 0**n

%o     if k == n: return n * T(n-1, n-1)

%o     return k * T(n-1, k-1) + (4*n - 4 + k) * T(n-1, k)

%o for n in range(8): print([T(n, k) for k in range(n + 1)])

%o # _Peter Luschny_, Mar 17 2024

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

%Y Cf. A371025.

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, Mar 08 2024