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%I #5 Apr 20 2016 08:36:07
%S 1,1,0,0,1,0,0,2,1,0,0,2,8,1,0,0,2,28,22,1,0,0,2,72,182,52,1,0,0,2,
%T 164,974,864,114,1,0,0,2,352,4174,8444,3474,240,1,0,0,2,732,15782,
%U 61464,57194,12660,494,1,0,0,2,1496,55286,373940,660842,332528,43358,1004,1,0
%N Triangle read by rows, T(n,k) = Sum_{j=0..n} C(-j,-n)*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>
%e Triangle starts:
%e 1,
%e 1, 0,
%e 0, 1, 0,
%e 0, 2, 1, 0,
%e 0, 2, 8, 1, 0,
%e 0, 2, 28, 22, 1, 0,
%e 0, 2, 72, 182, 52, 1, 0,
%e 0, 2, 164, 974, 864, 114, 1, 0
%p A271698 := (n,k) -> add(binomial(-j,-n)*combinat:-eulerian1(j,k), j=0..n):
%p seq(seq(A271698(n, k), k=0..n), n=0..10);
%t <<Combinatorica`
%t Flatten[Table[Sum[Binomial[-j,-n] Eulerian[j,k],{j,0,n}], {n,0,9},{k,0,n}]]
%Y A000255 (row sums), compare A028296 for alternating rows sums, A145654 and A005803 (diag. n,n-2).
%Y Cf. A173018.
%K nonn,tabl
%O 0,8
%A _Peter Luschny_, Apr 12 2016
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