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A271697 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.

%I A271697 #23 Nov 13 2024 22:30:02
%S A271697 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 A271697 113,813,813,113,1,0,0,1,239,3361,7631,3361,239,1,0,0,1,493,12421,
%U A271697 53833,53833,12421,493,1,0,0,1,1003,42865,320107,607009,320107,42865,1003,1,0
%N A271697 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 A271697 Andrew Howroyd, <a href="/A271697/b271697.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H A271697 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/ExtensionsOfTheBinomial">Extensions of the binomial</a>
%F A271697 T(n,k) = T(n,n-k). - _Alois P. Heinz_, Oct 29 2020
%e A271697 Triangle starts:
%e A271697   1;
%e A271697   0, 0;
%e A271697   0, 1,   0;
%e A271697   0, 1,   1,   0;
%e A271697   0, 1,   7,   1,   0;
%e A271697   0, 1,  21,  21,   1,   0;
%e A271697   0, 1,  51, 161,  51,   1, 0;
%e A271697   0, 1, 113, 813, 813, 113, 1, 0;
%e A271697   ...
%p A271697 A271697 := (n,k) -> add(binomial(-j-1,-n-1)*combinat:-eulerian1(j,k), j=0..n):
%p A271697 seq(seq(A271697(n, k), k=0..n), n=0..11);
%t A271697 <<Combinatorica`
%t A271697 Flatten[Table[Sum[Binomial[-j-1,-n-1] Eulerian[j,k],{j,0,n}], {n,0,9}, {k,0,n}]]
%t A271697 (* Second program (Combinatorica not needed): *)
%t A271697 E1[n_ /; n >= 0, 0] = 1;
%t A271697 E1[n_, k_] /; k < 0 || k > n = 0;
%t A271697 E1[n_, k_] := E1[n, k] = (n-k) E1[n-1, k-1] + (k+1) E1[n-1, k];
%t A271697 T[n_, k_] := Sum[Binomial[-j-1, -n-1] E1[j, k], {j, 0, n}];
%t A271697 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Oct 29 2020 *)
%o A271697 (PARI) T(n)={my(x='x+O('x^(n+1)), v=Vec(serlaplace((y-1)/(y*exp(x)-exp(x*y))))); vector(#v,n,Vecrev(v[n],n))}
%o A271697 { my(A=T(10)); for(i=1, #A, print(A[i])) } \\ _Andrew Howroyd_, Nov 13 2024
%Y A271697 Variant: A046739 (main entry for this triangle).
%Y A271697 Cf. A000166 (row sums), A122045 (Euler numbers are the alternating row sums), A070313 (col. 2) and (diag. n,n-2).
%Y A271697 Cf. A173018.
%Y A271697 T(2n,n) gives A320337.
%K A271697 nonn,tabl
%O A271697 0,13
%A A271697 _Peter Luschny_, Apr 12 2016