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.

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, 444, 720, 1, -14, 112, -350, 1099, 364, 3828, 5040, 1, -20, 210, -1064, 3969, -4340, 15980, 25584, 40320, 1, -27, 366, -2646, 12873, -31563, 79064, 34236, 270576, 362880

Offset: 0

Author

N. J. A. Sloane, Apr 17 2000

Keywords

• sign
• tabl
• nice
• easy

Comments

Apparently A190782 with reversed rows. - Mathew Englander, May 17 2014

Examples

			The first few polynomials are:
  1, 1+x, 2+x+x^2, 6+5*x+x^3, 24+14*x+11*x^2-2*x^3+x^4, ...
So the triangle begins:
  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,   444,   720;
  1, -14, 112,  -350, 1099,   364,  3828,  5040;
  1, -20, 210, -1064, 3969, -4340, 15980, 25584, 40320;
  ...
		

Links

• T. D. Noe, Rows n = 0..100 of triangle, flattened

Crossrefs

T(2*n,n) gives A347987.

Cf. A054649, A054655, A054654, A190782 .

Programs

Mathematica

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 *)

Formula

T(n, k) = Sum_{i=0..k} Stirling1(i+n-k,n-k)*n!/(i+n-k)!. - Igor Victorovich Statsenko, May 27 2024

Extensions

Missing 0 corrected by Steve Marak - N. J. A. Sloane, Jul 27 2012