A155123 A finite irregular triangle.

1, 2, 2, 2, 0, 4, 4, 0, -4, 8, 12, 0, 8, -32, 8, 48

Offset: 0

Author

Roger L. Bagula, Jan 20 2009

Keywords

• sign
• tabf
• fini
• full
• less

Examples

			The finite triangle, T(n,k), is:
  1;
  2;
  2,  2;
  0,  4,   4;
  0, -4,   8, 12;
  0,  8, -32,  8, 48;
		

Programs

Mathematica

t = {{1}, {1,1}, {1, 2*n, 1}, {1, -1 +2*n +2*n^2, -1 +2*n +2*n^2, 1}, {1, -2 +2*n +2*n^2 +2*n^3, 2 -8*n +4*n^2 +8*n^3, -2 +2*n +2*n^2 +2*n^3, 1}, {1, -3 +2*n +2*n^2 +2*n^3 +2*n^4, 2 +2*n -18*n^2 +2*n^3 +22*n^4, 2 +2*n - 18*n^2 +2*n^3 +22*n^4, -3 +2*n +2*n^2 +2*n^3 +2*n^4, 1}};
Table[CoefficientList[Apply[Plus, t[[m]]], n], {m,Length[t]}]//Flatten

Formula

T(n, k) = [x^k]( p(n, x) ), where p(0,x) = 1, p(1,x) = 2, p(2,x) = 2*x + 2, p(3,x) = 4*x^2 + 4*x, p(4,x) = 12*x^3 + 8*x^2 - 4*x, and p(5,x) = 48*x^4 + 8*x^3 - 32*x^2 + 8*x.

Extensions

Edited by G. C. Greubel, Jun 18 2024