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

0,2

Comments

Row sums of the triangle = A000712.

Links

Table of n, a(n) for n=0..45.

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.

Example

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.

Sequence in context: A325331 A333108 A266935 * A051693 A266470 A371745

Adjacent sequences: A175009 A175010 A175011 * A175013 A175014 A175015

Keyword

nonn,tabl,more

Author

Gary W. Adamson, Apr 03 2010

Status

approved