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.
1, 0, 1, 0, -4, 11, 0, 15, -156, 302, 0, -56, 1596, -9528, 15619, 0, 210, -14400, 193185, -882340, 1310354, 0, -792, 122265, -3213760, 30042672, -116857368, 162512286, 0, 3003, -1005004, 47887840, -802069632, 6034981134, -21078701112, 27971176092
Offset: 0
Examples
Triangle starts:
1;
0, 1;
0, -4, 11;
0, 15, -156, 302;
0, -56, 1596, -9528, 15619;
0, 210, -14400, 193185, -882340, 1310354;
0, -792, 122265, -3213760, 30042672, -116857368, 162512286;
Programs
-
Maple
T := (n, k) -> binomial(-k-n-1, -2*n-1)*combinat:-eulerian1(k+n, n): for n from 0 to 7 do seq(T(n,k), k=0..n) od;
-
Mathematica
E1[n_ /; n >= 0, 0] = 1; E1[n_, k_] /; k < 0 || k > n = 0; E1[n_, k_] := E1[n, k] = (n - k) E1[n - 1, k - 1] + (k + 1) E1[n - 1, k]; T[n_, k_] := Binomial[-k - n - 1, -2 n - 1] E1[n + k, n]; Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 30 2018 *)