A175012 Triangle generated from the g.f of A000712 (i.e., 1/(1-x^m)^2) interleaved with zeros.

1, 2, 2, 3, 2, 4, 4, 2, 4, 9, 5, 2, 4, 10, 14, 6, 2, 4, 10, 19, 23, 7, 2, 4, 10, 20, 34, 32, 8, 2, 4, 10, 20, 39, 55, 46, 9, 2, 4, 10, 20, 40, 66, 88, 60, 10

Offset: 0

Author

Gary W. Adamson, Apr 03 2010

Keywords

• nonn
• tabl
• more

Comments

Row sums of the triangle = A000712.

Examples

			First few rows of the array =
  1, 2, 3,  4,  5,  6,  7,   8,   9,  10, ...
  1, 2, 5,  8, 14, 20, 30,  40,  55,  70, ...
  1, 2, 5, 10, 18, 30, 49,  74, 110, 158, ...
  1, 2, 5, 10, 20, 34, 59,  94, 149, 224, ...
  1, 2, 5, 10, 20, 36, 63, 104, 169, 264, ...
  1, 2, 5, 10, 20, 36, 65, 108, 179, 284, ...
  ...
First few rows of the triangle =
  1;
  2;
  2,  3;
  2,  4,  4;
  2,  4,  9,  5;
  2,  4, 10, 14,  6;
  2,  4, 10, 19, 23,  7;
  2,  4, 10, 20, 34, 32,  8;
  2,  4, 10, 20, 39, 55, 46,  9;
  2,  4, 10, 20, 40, 66, 88, 60, 10;
  ...
		

Crossrefs

Cf. A000712.

Formula

Given 1/(1-x^m)^2 = S(x) = (1 + 2x + 3x^2 + ...), let a = S(x), b = S(x^2) (i.e., S(x) interleaved with one zero); S(x^3) = S(x) interleaved with two zeros = c, etc.; then row 1 = a, row 2 = a*b, row 3 = a*b*c, ...

Take finite differences of the array from the top down, becoming rows of the triangle.