A124732 Triangle P*M, where P is the Pascal triangle written as an infinite lower triangular matrix and M is the infinite bidiagonal matrix with (1,2,1,2,...) in the main diagonal and (2,1,2,1,...) in the subdiagonal.

1, 3, 2, 5, 5, 1, 7, 9, 5, 2, 9, 14, 14, 9, 1, 11, 20, 30, 25, 7, 2, 13, 27, 55, 55, 27, 13, 1, 15, 35, 91, 105, 77, 49, 9, 2, 17, 44, 140, 182, 182, 140, 44, 17, 1, 19, 54, 204, 294, 378, 336, 156, 81, 11, 2, 21, 65, 285, 450, 714, 714, 450, 285, 65, 21, 1, 23, 77, 385, 660

Offset: 1

Author

Gary W. Adamson & Roger L. Bagula, Nov 05 2006

Keywords

• nonn
• tabl

Comments

Row sums = A052940: (1, 5, 11, 23, 47, 95, ...).

Examples

			First 3 rows of the triangle are (1; 3,2; 5,5,1) since [1,0,0; 1,1,0; 1,2,1] * [1,0,0; 2,2,0; 0,1,1] = [1,0,0; 3,2,0; 5,5,1].
First few rows of the triangle are:
   1;
   3,   2;
   5,   5,   1;
   7,   9,   5,   2;
   9,  14,  14,   9,   1;
  11,  20,  30,  25,   7,   2;
  13,  27,  55,  55,  27,  13,   1;
  15,  35,  91, 105,  77,  49,   9,   2;
  ...
		

Crossrefs

Cf. A124730, A052940.

Programs

Maple

T:=(n,k)->binomial(n,k)*(3*n-(-1)^k*(n-2*k))/2/n: for n from 1 to 12 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

Formula

T(n,k) = binomial(n,k)*(3n-(-1)^k*(n-2*k))/(2n) (1 <= k <= n).

Extensions

Edited by N. J. A. Sloane, Nov 24 2006