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

Author

Peter Luschny, Dec 18 2018

Keywords

• sign
• tabl

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;
		

Crossrefs

Row sums give A320337.

Cf. A046739, A180056 (main diagonal), A271697, A001791.

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 *)