cp's OEIS Frontend

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

%I A371025 #13 Mar 17 2024 18:23:15
%S A371025 1,0,1,0,3,2,0,15,18,6,0,105,174,108,24,0,945,1950,1710,720,120,0,
%T A371025 10395,25290,28080,16920,5400,720,0,135135,374850,497070,383040,
%U A371025 176400,45360,5040,0,2027025,6267870,9574740,8883000,5266800,1965600,423360,40320
%N A371025 Triangle read by rows: T(n, k) = 2^n*Sum_{j=0..k} (-1)^(k - j)*binomial(k, j)* Pochhammer(j/2, n).
%F A371025 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
%e A371025 Triangle starts:
%e A371025 [0] 1;
%e A371025 [1] 0,      1;
%e A371025 [2] 0,      3,      2;
%e A371025 [3] 0,     15,     18,      6;
%e A371025 [4] 0,    105,    174,    108,     24;
%e A371025 [5] 0,    945,   1950,   1710,    720,    120;
%e A371025 [6] 0,  10395,  25290,  28080,  16920,   5400,   720;
%e A371025 [7] 0, 135135, 374850, 497070, 383040, 176400, 45360, 5040;
%p A371025 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);
%o A371025 (SageMath)
%o A371025 from functools import cache
%o A371025 @cache
%o A371025 def T(n, k):   # after _Werner Schulte_
%o A371025     if k == 0: return 0**n
%o A371025     if k == n: return n * T(n-1, n-1)
%o A371025     return k * T(n-1, k-1) + (2*n - 2 + k) * T(n-1, k)
%o A371025 for n in range(8): print([T(n, k) for k in range(n + 1)])
%o A371025 # _Peter Luschny_, Mar 17 2024
%Y A371025 Cf. A000142 (main diagonal), A001147 (column 1), A308939 (row sums).
%K A371025 nonn,tabl
%O A371025 0,5
%A A371025 _Peter Luschny_, Mar 08 2024