A178616 Triangle by columns, odd columns of Pascal's triangle A007318, otherwise (1, 0, 0, 0, ...).

1, 0, 1, 0, 2, 1, 0, 3, 0, 1, 0, 4, 0, 4, 1, 0, 5, 0, 10, 0, 1, 0, 6, 0, 20, 0, 6, 1, 0, 7, 0, 35, 0, 21, 0, 1, 0, 8, 0, 56, 0, 56, 0, 8, 0, 1, 0, 9, 0, 84, 0, 126, 0, 36, 0, 1, 0, 10, 0, 120, 0, 252, 0, 120, 0, 10, 1

Offset: 0

Author

Gary W. Adamson, May 30 2010

Keywords

• nonn
• tabl

Comments

Row sums = a variant of A052950, starting (1, 1, 3, 4, 9, 16, 33, ...); whereas A052950 starts (2, 1, 3, 4, 9, ...).

Column 1 of the inverse of A178616 is a signed variant of A065619 prefaced with a 0; where A065619 = (1, 2, 3, 8, 25, 96, 427, ...).

Examples

			First few rows of the triangle:
  1,
  0,  1;
  0,  2, 1;
  0,  3, 0,   1
  0,  4, 0,   4, 1;
  0,  5, 0,  10, 0,   1;
  0,  6, 0,  20, 0,   6, 1;
  0,  7, 0,  35, 0,  21, 0,   1;
  0,  8, 0,  56, 0,  56, 0,   8, 1;
  0,  9, 0,  84, 0, 126, 0,  36, 0,  1;
  0, 10, 0, 120, 0, 252, 0, 120, 0, 10, 1;
  0, 11, 0, 165, 0, 462, 0, 330, 0, 55, 0, 1;
  ...
		

Crossrefs

Cf. A162109, A065619, A052950, A095704.

Formula

Triangle, odd columns of Pascal's triangle; (1, 0, 0, 0, ...) as even columns k.

Alternatively, (since A178616 + A162169 - Identity matrix) = Pascal's triangle,

we can begin with Pascal's triangle, subtract A162169, then add the Identity

matrix to obtain A178616.