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A321967 Triangle read by rows, T(n,k) = binomial(-k-n-1, -2*n-1)*E1(k+n, n), E1 the Eulerian numbers A173018, for n >= 0 and 0 <= k <= n.

%I A321967 #19 Dec 31 2018 04:13:58
%S A321967 1,0,1,0,-4,11,0,15,-156,302,0,-56,1596,-9528,15619,0,210,-14400,
%T A321967 193185,-882340,1310354,0,-792,122265,-3213760,30042672,-116857368,
%U A321967 162512286,0,3003,-1005004,47887840,-802069632,6034981134,-21078701112,27971176092
%N A321967 Triangle read by rows, T(n,k) = binomial(-k-n-1, -2*n-1)*E1(k+n, n), E1 the Eulerian numbers A173018, for n >= 0 and 0 <= k <= n.
%e A321967 Triangle starts:
%e A321967                        1;
%e A321967                   0,        1;
%e A321967               0,      -4,       11;
%e A321967           0,     15,      -156,      302;
%e A321967        0,   -56,     1596,    -9528,     15619;
%e A321967     0,   210,  -14400,   193185,   -882340,   1310354;
%e A321967   0, -792, 122265, -3213760, 30042672, -116857368, 162512286;
%p A321967 T := (n, k) -> binomial(-k-n-1, -2*n-1)*combinat:-eulerian1(k+n, n):
%p A321967 for n from 0 to 7 do seq(T(n,k), k=0..n) od;
%t A321967 E1[n_ /; n >= 0, 0] = 1; E1[n_, k_] /; k < 0 || k > n = 0;
%t A321967 E1[n_, k_] := E1[n, k] = (n - k) E1[n - 1, k - 1] + (k + 1) E1[n - 1, k];
%t A321967 T[n_, k_] := Binomial[-k - n - 1, -2 n - 1] E1[n + k, n];
%t A321967 Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten
%t A321967 (* _Jean-François Alcover_, Dec 30 2018 *)
%Y A321967 Row sums give A320337.
%Y A321967 Cf. A046739, A180056 (main diagonal), A271697, A001791.
%K A321967 sign,tabl
%O A321967 0,5
%A A321967 _Peter Luschny_, Dec 18 2018