A291977 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.
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, 408, -136, 0, 1, 113, 492, -1774, 1072, 912, -1088, 272, 1, 239, 2592, -5008, -6656, 20736, -15872, 3968, 0, 1, 493, 10628, -50, -94432, 154528, -57856, -45056, 39680, -7936
Offset: 0
Examples
Triangle starts: 0| 1 1| 1, -1 2| 1, -1, 0 3| 1, 1, -4, 2 4| 1, 7, -16, 8, 0 5| 1, 21, -28, -26, 48, -16 6| 1, 51, 32, -356, 408, -136, 0 7| 1, 113, 492, -1774, 1072, 912, -1088, 272 8| 1, 239, 2592, -5008, -6656, 20736, -15872, 3968, 0 9| 1, 493, 10628, -50, -94432, 154528, -57856, -45056, 39680, -7936 --------------------------------------------------------------------- k| 0 1 2 3 4 5 6 7 8 9
Programs
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Maple
with(combinat): T := (n, k) -> add((-1)^(k-j)*eulerian1(n, j)*binomial(n-j, n-k), j=0..n): seq(print(seq(T(n, k), k=0..n)), n=0..9);
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Python
from sympy.core.cache import cacheit from sympy import binomial @cacheit 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) def T(n, k): return sum([(-1)**(k - j)*eulerian1(n, j)*binomial(n - j, n - k) for j in range(n + 1)]) for n in range(10): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Sep 11 2017