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Triangle read by rows, T(n,k) = Sum_{j=0..n} C(-j-1,-n-1)*E1(j,k), E1 the Eulerian numbers A173018, for n>=0 and 0<=k<=n.
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%I #19 Oct 30 2020 14:58:59

%S 1,0,0,0,1,0,0,1,1,0,0,1,7,1,0,0,1,21,21,1,0,0,1,51,161,51,1,0,0,1,

%T 113,813,813,113,1,0,0,1,239,3361,7631,3361,239,1,0,0,1,493,12421,

%U 53833,53833,12421,493,1,0,0,1,1003,42865,320107,607009,320107,42865,1003,1,0

%N Triangle read by rows, T(n,k) = Sum_{j=0..n} C(-j-1,-n-1)*E1(j,k), E1 the Eulerian numbers A173018, for n>=0 and 0<=k<=n.

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/ExtensionsOfTheBinomial">Extensions of the binomial</a>

%F T(n,k) = T(n,n-k). - _Alois P. Heinz_, Oct 29 2020

%e Triangle starts:

%e 1;

%e 0, 0;

%e 0, 1, 0;

%e 0, 1, 1, 0;

%e 0, 1, 7, 1, 0;

%e 0, 1, 21, 21, 1, 0;

%e 0, 1, 51, 161, 51, 1, 0;

%e 0, 1, 113, 813, 813, 113, 1, 0;

%e ...

%p A271697 := (n,k) -> add(binomial(-j-1,-n-1)*combinat:-eulerian1(j,k), j=0..n):

%p seq(seq(A271697(n, k), k=0..n), n=0..11);

%t <<Combinatorica`

%t Flatten[Table[Sum[Binomial[-j-1,-n-1] Eulerian[j,k],{j,0,n}], {n,0,9}, {k,0,n}]]

%t (* Second program (Combinatorica not needed): *)

%t E1[n_ /; n >= 0, 0] = 1;

%t E1[n_, k_] /; k < 0 || k > n = 0;

%t E1[n_, k_] := E1[n, k] = (n-k) E1[n-1, k-1] + (k+1) E1[n-1, k];

%t T[n_, k_] := Sum[Binomial[-j-1, -n-1] E1[j, k], {j, 0, n}];

%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Oct 29 2020 *)

%Y Variant: A046739 (main entry for this triangle).

%Y Cf. A000166 (row sums), A122045 (Euler numbers are the alternating row sums), A070313 (col. 2) and (diag. n,n-2).

%Y Cf. A173018.

%Y T(2n,n) gives A320337.

%K nonn,tabl

%O 0,13

%A _Peter Luschny_, Apr 12 2016