A137854 Triangle generated from an array: A008277 * A008277(transform).

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 11, 8, 1, 1, 16, 28, 28, 16, 1, 1, 32, 71, 87, 71, 32, 1, 1, 64, 184, 266, 266, 184, 64, 1, 1, 128, 491, 823, 952, 823, 491, 128, 1, 1, 256, 1348, 2598, 3381, 381, 2598, 1348, 2561

Offset: 1

Author

Gary W. Adamson, Feb 15 2008

Keywords

• nonn
• tabl

Comments

Row sums = A000995 such that row 1 = A000995(3) = 1.

This array is the product of the lower triangular Stirling matrix and its transpose, which explains why the array is symmetric. - David Callan, Dec 02 2011

In the triangle, T(n,k) is the number of permutations of [n+1] that avoid both dashed patterns 1-23 and 3-12, start with an ascent, and have first entry k. For example, T(4,2)=4 counts 23154, 24153, 24315, 25431. - David Callan, Dec 02 2011

Examples

			First few rows of the array:
  1,  1,  1,   1,   1,    1, ...
  1,  2,  4,   8,  16,   32, ...
  1,  4, 11,  28,  71,  184, ...
  1,  8, 28,  87, 266,  823, ...
  1, 16, 71, 266, 952, 3381, ...
  ...
First few rows of the triangle:
  1;
  1,   1;
  1,   2,   1;
  1,   4,   4,   1;
  1,   8,  11,   8,   1;
  1,  16,  28,  28,  16,   1;
  1,  32,  71,  87,  71,  32,   1;
  1,  64, 184, 266, 266, 184,  64,   1;
  1, 128, 491, 823, 952, 823, 491, 128,   1;
  ...
		

Crossrefs

Cf. A000995, A008277.

Formula

Triangle read by rows = antidiagonals of an array formed by A008277 * A008277(transform), where A008277 = the Stirling number of the second kind triangle.