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A271697
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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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4
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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, 113, 813, 813, 113, 1, 0, 0, 1, 239, 3361, 7631, 3361, 239, 1, 0, 0, 1, 493, 12421, 53833, 53833, 12421, 493, 1, 0, 0, 1, 1003, 42865, 320107, 607009, 320107, 42865, 1003, 1, 0
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OFFSET
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0,13
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LINKS
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FORMULA
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EXAMPLE
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Triangle starts:
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, 113, 813, 813, 113, 1, 0;
...
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MAPLE
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A271697 := (n, k) -> add(binomial(-j-1, -n-1)*combinat:-eulerian1(j, k), j=0..n):
seq(seq(A271697(n, k), k=0..n), n=0..11);
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MATHEMATICA
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<<Combinatorica`
Flatten[Table[Sum[Binomial[-j-1, -n-1] Eulerian[j, k], {j, 0, n}], {n, 0, 9}, {k, 0, n}]]
(* Second program (Combinatorica not needed): *)
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_] := Sum[Binomial[-j-1, -n-1] E1[j, k], {j, 0, n}];
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CROSSREFS
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Variant: A046739 (main entry for this triangle).
Cf. A000166 (row sums), A122045 (Euler numbers are the alternating row sums), A070313 (col. 2) and (diag. n,n-2).
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KEYWORD
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AUTHOR
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STATUS
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approved
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