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Triangle T(n,k)= n! if k=0, T(n,k) = -(n-k)!*A003319(k) if k > 0.
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%I #14 Feb 19 2018 03:34:50

%S 1,1,-1,2,-1,-1,6,-2,-1,-3,24,-6,-2,-3,-13,120,-24,-6,-6,-13,-71,720,

%T -120,-24,-18,-26,-71,-461,5040,-720,-120,-72,-78,-142,-461,-3447,

%U 40320,-5040,-720,-360,-312,-426,-922,-3447,-29093,362880,-40320,-5040,-2160,-1560

%N Triangle T(n,k)= n! if k=0, T(n,k) = -(n-k)!*A003319(k) if k > 0.

%F T(n,0) = A000142(n).

%F T(n,k) = -A141476(n-1,k-1), k > 0.

%F Sum_{k=0..n} |T(n,k)| = 2*n! = A098558(n).

%e Triangle begins

%e 1;

%e 1, -1;

%e 2, -1, -1;

%e 6, -2, -1, -3;

%e 24, -6, -2, -3, -13;

%e 120, -24, -6, -6, -13, -71;

%e 720, -120, -24, -18, -26, -71, -461;

%t (* b = A003319 *) b[0]=0; b[n_] := b[n] = n! - Sum[k! b[n-k], {k, 1, n-1}]; T[n_, 0] := n!; T[n_, k_] := -(n - k)! b[k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Feb 18 2018 *)

%K sign,tabl

%O 0,4

%A _Paul Curtz_, Sep 16 2008