A124216 Generalized Pascal triangle.

1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 16, 34, 16, 1, 1, 25, 90, 90, 25, 1, 1, 36, 195, 328, 195, 36, 1, 1, 49, 371, 931, 931, 371, 49, 1, 1, 64, 644, 2240, 3334, 2240, 644, 64, 1, 1, 81, 1044, 4788, 9846, 9846

Offset

0,5

Comments

Consider the 1-parameter family of triangles with g.f. (1-x(1+y))/(1-2x(1+y)+x^2(1+k*x+y^2)). A007318 corresponds to k=2. A056241 corresponds to k=1. This sequence corresponds to k=0. Row sums are A006012. Diagonal sums are A124217.

Links

Table of n, a(n) for n=0..50.

Paul Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011) # 11.4.5.

Formula

G.f.: (1-x(1+y))/(1-2x(1+y)+x^2(1+y^2)); Number triangle T(n,k)=sum{j=0..n, C(n,j)C(j,2(j-k))2^(j-k)}.

Equals 2*A001263 - A007318; (i.e. twice the Narayana triangle minus Pascal's triangle). - Gary W. Adamson, Jun 14 2007

Example

Triangle begins
  1,
  1, 1,
  1, 4, 1,
  1, 9, 9, 1,
  1, 16, 34, 16, 1,
  1, 25, 90, 90, 25, 1,
  1, 36, 195, 328, 195, 36, 1,
  1, 49, 371, 931, 931, 371, 49, 1

Crossrefs

Cf. A001263, A007318, A056241, A006012, A124217.

Sequence in context: A082043 A177944 A174006 * A008459 A259333 A180960

Adjacent sequences: A124213 A124214 A124215 * A124217 A124218 A124219

Keyword

easy,nonn,tabl

Author

Paul Barry, Oct 19 2006

Status

approved