A173305 Triangle by columns, A000009 in every column shifted down twice for k > 0.

1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 2, 1, 4, 2, 1, 1, 5, 3, 2, 1, 6, 4, 2, 1, 1, 8, 5, 3, 2, 1, 10, 6, 4, 2, 1, 1, 12, 8, 5, 3, 2, 1, 15, 10, 6, 4, 2, 1, 1, 18, 12, 8, 5, 3, 2, 1, 22, 15, 10, 6, 4, 2, 1, 1, 27, 18, 12, 8, 5, 3, 2, 1, 32, 22, 15, 10, 6, 4, 2, 1, 1

Offset: 0

Author

Gary W. Adamson, Feb 15 2010

Keywords

• nonn
• tabf

Comments

Row sums = A038348.

Let the triangle = M. Limit_{n->oo} M^n = the partition numbers, A000041;

equivalent to the statement A000009(x) = A000041(x)/A000041(x^2), or

(1 + x + x^2 + 2x^3 + 2x^4 +3x^5 + 4x^6 + ...) = (1 + x + 2x^2 + 3x^3 + ...)/(1 + x^2 + 2x^4 + 3x^6 + 5x^8 + 7x^10 + ...).

Examples

			Triangle begins:
   1;
   1;
   1,  1;
   2,  1;
   2,  1,  1;
   3,  2,  1;
   4,  2,  1,  1;
   5,  3,  2,  1;
   6,  4,  2,  1,  1;
   8,  5,  3,  2,  1;
  10,  6,  4,  2,  1,  1;
  12,  8,  5,  3,  2,  1;
  15, 10,  6,  4,  2,  1,  1;
  18, 12,  8,  5,  3,  2,  1;
  22, 15, 10,  6,  4,  2,  1,  1;
  27, 18, 12,  8,  5,  3,  2,  1;
  32, 22, 15, 10,  6,  4,  2,  1,  1;
  38, 27, 18, 12,  8,  5,  3,  2,  1;
  46, 32, 22, 15, 10,  6,  4,  2,  1,  1;
  54, 38, 27, 18, 12,  8,  5,  3,  2,  1;
  ...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10301 (rows 0..200 of the triangle, flattened).

Crossrefs

Cf. A173306, A038348, A000009, A000041.

Programs

Mathematica

Table[PartitionsQ[n-2*k], {n, 0, 15}, {k, 0, n/2}] (* Paolo Xausa, Feb 21 2024 *)

Formula

Triangle by columns, A000009 in every column shifted down twice for k > 0.

T(n,k) = A000009(n-2*k). - Paolo Xausa, Feb 21 2024