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

0,5

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, ...).

Links

Table of n, a(n) for n=0..66.

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.

Example

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.

Sequence in context: A157409 A245960 A340867 * A165252 A371954 A127373

Adjacent sequences: A178613 A178614 A178615 * A178617 A178618 A178619

Keyword

nonn,tabl

Author

Gary W. Adamson, May 30 2010

Status

approved