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%I #19 Oct 19 2023 16:28:18
%S 1,2,0,4,1,0,8,7,1,0,16,33,15,1,0,32,131,131,31,1,0,64,473,883,473,63,
%T 1,0,128,1611,5111,5111,1611,127,1,0,256,5281,26799,44929,26799,5281,
%U 255,1,0,512,16867,131275,344551,344551,131275,16867,511,1,0
%N Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*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 E.g.f.: (exp(x)*(y - 1))/(y - exp(x*(y - 1))). - _Peter Luschny_, Aug 14 2022
%F T(n,k) = Sum_{i=0..n} Binomial(n,i)*Eulerian(i,k), where Eulerian(n,k) = Eulerian numbers A173018. Equivalently, if T is the matrix generated by T(n,k), B is the binomial matrix and E is the Eulerian matrix, then T = B E. - _Emanuele Munarini_, Oct 19 2023
%e Triangle starts:
%e [1]
%e [2, 0]
%e [4, 1, 0]
%e [8, 7, 1, 0]
%e [16, 33, 15, 1, 0]
%e [32, 131, 131, 31, 1, 0]
%e [64, 473, 883, 473, 63, 1, 0]
%e [128, 1611, 5111, 5111, 1611, 127, 1, 0]
%p T := (n, k) -> add((-1)^(n-j)*combinat:-eulerian1(j,k)*binomial(-j-1,-n-1), j=0..n): seq(seq(T(n, k), k=0..n), n=0..10);
%p # Or:
%p egf := (exp(x)*(y - 1))/(y - exp(x*(y - 1))); ser := series(egf, x, 12):
%p cx := n -> series(coeff(ser, x, n), y, n + 2):
%p seq(seq(n!*coeff(cx(n), y, k), k = 0..n), n = 0..9); # _Peter Luschny_, Aug 14 2022
%t <<Combinatorica`
%t Flatten[Table[Sum[(-1)^(n-j) Binomial[-j-1,-n-1] Eulerian[j,k], {j,0,n}], {n,0,9},{k,0,n}]]
%t Flatten[Table[Sum[Binomial[n,i] Eulerian[i,k], {i,0,n}], {n,0,12}, {k,0,n}]] (* _Emanuele Munarini_, Oct 19 2023 *)
%Y Cf. A000522 (row sums), A000079 (col. 0), A066810 (col. 1).
%Y Cf. A173018.
%K nonn,tabl
%O 0,2
%A _Peter Luschny_, Apr 20 2016
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