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%I #14 Sep 11 2017 19:49:09
%S 1,1,-1,1,-1,0,1,1,-4,2,1,7,-16,8,0,1,21,-28,-26,48,-16,1,51,32,-356,
%T 408,-136,0,1,113,492,-1774,1072,912,-1088,272,1,239,2592,-5008,-6656,
%U 20736,-15872,3968,0,1,493,10628,-50,-94432,154528,-57856,-45056,39680,-7936
%N Triangle read by rows, T(n, k) = Sum_{j=0..n} (-1)^(k-j)*Eulerian1(n, j)* binomial(n-j, n-k) for 0 <= k <= n.
%F T(n, k) = Sum_{j=0..n} (-1)^(k-j)*A173018(n, j)*A007318(n-j, n-k) for 0 <= k <= n.
%e Triangle starts:
%e 0| 1
%e 1| 1, -1
%e 2| 1, -1, 0
%e 3| 1, 1, -4, 2
%e 4| 1, 7, -16, 8, 0
%e 5| 1, 21, -28, -26, 48, -16
%e 6| 1, 51, 32, -356, 408, -136, 0
%e 7| 1, 113, 492, -1774, 1072, 912, -1088, 272
%e 8| 1, 239, 2592, -5008, -6656, 20736, -15872, 3968, 0
%e 9| 1, 493, 10628, -50, -94432, 154528, -57856, -45056, 39680, -7936
%e ---------------------------------------------------------------------
%e k| 0 1 2 3 4 5 6 7 8 9
%p with(combinat):
%p T := (n, k) -> add((-1)^(k-j)*eulerian1(n, j)*binomial(n-j, n-k), j=0..n):
%p seq(print(seq(T(n, k), k=0..n)), n=0..9);
%o (Python)
%o from sympy.core.cache import cacheit
%o from sympy import binomial
%o @cacheit
%o def eulerian1(n, k): return 1 if k==0 else 0 if k==n else eulerian1(n - 1, k)*(k + 1) + eulerian1(n - 1, k - 1)*(n - k)
%o def T(n, k): return sum([(-1)**(k - j)*eulerian1(n, j)*binomial(n - j, n - k) for j in range(n + 1)])
%o for n in range(10): print([T(n, k) for k in range(n + 1)]) # _Indranil Ghosh_, Sep 11 2017
%Y Cf. A007318, A142073, A173018.
%K sign,tabl
%O 0,9
%A _Peter Luschny_, Sep 10 2017