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A375855 Triangle read by rows: T(n, k) = 2^k * hypergeom([-n, -k], [], -1/2).

%I A375855 #18 Sep 09 2024 20:00:40
%S A375855 1,1,1,1,0,-2,1,-1,-2,2,1,-2,0,8,8,1,-3,4,8,-24,-88,1,-4,10,-4,-56,32,
%T A375855 592,1,-5,18,-34,-40,312,400,-3344,1,-6,28,-88,96,512,-1472,-6144,
%U A375855 14464,1,-7,40,-172,448,32,-4544,4160,63616,-2944
%N A375855 Triangle read by rows: T(n, k) = 2^k * hypergeom([-n, -k], [], -1/2).
%H A375855 John Tyler Rascoe, <a href="/A375855/b375855.txt">Rows n = 0..140, flattened</a>
%F A375855 T(n, k) = (-1)^k*Sum_{j=0..k} (-2)^(k - j)*binomial(n, j)*binomial(k, j)*j!.
%e A375855 Triangle starts:
%e A375855 [0] 1;
%e A375855 [1] 1,  1;
%e A375855 [2] 1,  0, -2;
%e A375855 [3] 1, -1, -2,    2;
%e A375855 [4] 1, -2,  0,    8,   8;
%e A375855 [5] 1, -3,  4,    8, -24, -88;
%e A375855 [6] 1, -4, 10,   -4, -56,  32,   592;
%e A375855 [7] 1, -5, 18,  -34, -40, 312,   400, -3344;
%e A375855 [8] 1, -6, 28,  -88,  96, 512, -1472, -6144, 14464;
%e A375855 [9] 1, -7, 40, -172, 448,  32, -4544,  4160, 63616, -2944;
%e A375855 ...
%p A375855 T := (n, k) -> 2^k * hypergeom([-n, -k], [], -1/2);
%p A375855 for n from 0 to 9 do seq(simplify(T(n, k)), k=0..n) od; # _Peter Luschny_, Sep 02 2024
%t A375855 T[n_, k_] := (-1)^k*Sum[(-2)^(k - j)*Binomial[n, j]*Binomial[k, j]*j!, {j, 0, k}];
%t A375855 Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
%o A375855 (Python)
%o A375855 from math import comb, factorial
%o A375855 def A375855(n,k):
%o A375855     return (-1)**k*sum((-2)**(k-j)*comb(n, j)*comb(k, j)*factorial(j) for j in range(k+1)) # _John Tyler Rascoe_, Sep 05 2024
%Y A375855 Cf. A375854, A000012, A295382 (main diagonal).
%K A375855 sign,tabl,easy
%O A375855 0,6
%A A375855 _Detlef Meya_, Aug 31 2024