A168296 Worpitzky form polynomials for the {1,16,1} A142462 sequence: p(x,n) = Sum_{k=1..n} A(n, k)*binomial(x + k - 1, n - 1).

1, 1, 2, 2, 18, 18, 6, 156, 432, 288, 24, 792, 7416, 13248, 6624, 120, -11280, 64800, 374400, 496800, 198720, 720, -62640, -1254960, 4968000, 20865600, 22057920, 7352640, 5040, 24012000, -11854080, -125677440, 389491200, 1288103040, 1132306560, 323516160

Offset: 1

Author

Roger L. Bagula, Nov 22 2009

Keywords

• uned
• sign
• tabl

Examples

			Triangle begins:
  {1},
  {1, 2},
  {2, 18, 18},
  {6, 156, 432, 288},
  {24, 792, 7416, 13248, 6624},
  {120, -11280, 64800, 374400, 496800, 198720},
  {720, -62640, -1254960, 4968000, 20865600, 22057920, 7352640},
  {5040, 24012000, -11854080, -125677440, 389491200, 1288103040, 1132306560, 323516160},
  ...
		

Crossrefs

Cf. A142462.

Programs

Mathematica

(* Worpitzky form polynomials for A142462 *)
m = 7;
A[n_, 1] := 1 A[n_, n_] := 1
A[n_, k_] := (m*n - m*k + 1)A[n - 1, k - 1] + (m*k - (m - 1))A[n - 1, k];
a = Table[A[n, k], {n, 10}, {k, n}];
p[x_, n_] = Sum[a[[n, k]]*Binomial[x + k - 1, n - 1], {k, 1, n}];
Table[CoefficientList[Expand[(n - 1)!*p[x, n]], x], {n, 1, 10}];
Flatten[%]

Formula

p(x,n) = Sum_{k=1..n} A(n, k)*binomial(x + k - 1, n - 1).