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A054651 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.

%I A054651 #43 May 27 2024 09:18:22
%S A054651 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 A054651 444,720,1,-14,112,-350,1099,364,3828,5040,1,-20,210,-1064,3969,-4340,
%U A054651 15980,25584,40320,1,-27,366,-2646,12873,-31563,79064,34236,270576,362880
%N A054651 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 A054651 Apparently A190782 with reversed rows. - _Mathew Englander_, May 17 2014
%H A054651 T. D. Noe, <a href="/A054651/b054651.txt">Rows n = 0..100 of triangle, flattened</a>
%F A054651 T(n, k) = Sum_{i=0..k} Stirling1(i+n-k,n-k)*n!/(i+n-k)!. - _Igor Victorovich Statsenko_, May 27 2024
%e A054651 The first few polynomials are:
%e A054651   1, 1+x, 2+x+x^2, 6+5*x+x^3, 24+14*x+11*x^2-2*x^3+x^4, ...
%e A054651 So the triangle begins:
%e A054651   1;
%e A054651   1,   1;
%e A054651   1,   1,   2;
%e A054651   1,   0,   5,     6;
%e A054651   1,  -2,  11,    14,   24;
%e A054651   1,  -5,  25,     5,   94,   120;
%e A054651   1,  -9,  55,   -75,  304,   444,   720;
%e A054651   1, -14, 112,  -350, 1099,   364,  3828,  5040;
%e A054651   1, -20, 210, -1064, 3969, -4340, 15980, 25584, 40320;
%e A054651   ...
%t A054651 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 A054651 T(2*n,n) gives A347987.
%Y A054651 Cf. A054649, A054655, A054654, A190782 .
%K A054651 sign,tabl,nice,easy
%O A054651 0,6
%A A054651 _N. J. A. Sloane_, Apr 17 2000
%E A054651 Missing 0 corrected by Steve Marak - _N. J. A. Sloane_, Jul 27 2012