A321967 - OEIS

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

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OFFSET

0,5

LINKS

Table of n, a(n) for n=0..35.

EXAMPLE

Triangle starts:

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;

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 *)

CROSSREFS

Row sums give A320337.

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

Sequence in context: A356110 A048873 A091390 * A230024 A132150 A091389

Adjacent sequences: A321964 A321965 A321966 * A321968 A321969 A321970

KEYWORD

sign,tabl

AUTHOR

Peter Luschny, Dec 18 2018

STATUS

approved