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%I #37 Sep 27 2021 07:57:21
%S 1,1,1,1,1,2,1,0,5,6,1,-2,11,14,24,1,-5,25,5,94,120,1,-9,55,-75,304,
%T 444,720,1,-14,112,-350,1099,364,3828,5040,1,-20,210,-1064,3969,-4340,
%U 15980,25584,40320,1,-27,366,-2646,12873,-31563,79064,34236,270576,362880
%N Triangle T(n,k) read by rows giving coefficients in expansion of n! * Sum_{i=0..n} C(x,i) in descending powers of x.
%C Apparently A190782 with reversed rows. - _Mathew Englander_, May 17 2014
%H T. D. Noe, <a href="/A054651/b054651.txt">Rows n = 0..100 of triangle, flattened</a>
%e The first few polynomials are:
%e 1, 1+x, 2+x+x^2, 6+5*x+x^3, 24+14*x+11*x^2-2*x^3+x^4, ...
%e So the triangle begins:
%e 1;
%e 1, 1;
%e 1, 1, 2;
%e 1, 0, 5, 6;
%e 1, -2, 11, 14, 24;
%e 1, -5, 25, 5, 94, 120;
%e 1, -9, 55, -75, 304, 444, 720;
%e 1, -14, 112, -350, 1099, 364, 3828, 5040;
%e 1, -20, 210, -1064, 3969, -4340, 15980, 25584, 40320;
%e ...
%t c[n_, k_] := Product[n-i, {i, 0, k-1}]/k!; row[n_] := CoefficientList[ n!*Sum[c[x, k], {k, 0, n}], x] // Reverse; Table[ row[n], {n, 0, 9}] // Flatten (* _Jean-François Alcover_, Oct 04 2012 *)
%Y T(2*n,n) gives A347987.
%Y Cf. A054649, A054655, A054654.
%K sign,tabl,nice,easy
%O 0,6
%A _N. J. A. Sloane_, Apr 17 2000
%E Missing 0 corrected by Steve Marak - _N. J. A. Sloane_, Jul 27 2012
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