A067327 Triangle related to generalized Catalan numbers A064340.

1, 1, 3, 4, 12, 12, 28, 84, 96, 48, 256, 768, 912, 576, 192, 2704, 8112, 9792, 6720, 3072, 768, 31168, 93504, 113856, 81408, 42240, 15360, 3072, 380608, 1141824, 1397760, 1023744, 568320, 242688, 73728

Offset: 0

Author

Wolfdieter Lang, Feb 05 2002

Keywords

• nonn
• easy
• tabl

Comments

The row polynomials Z(2,2; n,y)= sum(a(n,m)*y^m,m=0..n) appear in c(2,2; x) (the g.f. of C(2,2; n) := A064340(n)) with the first (n+1) expansion terms subtracted, as follows: c(2,2; x)-sum(C(2,2; k)*x^k,k=0..n) = x^(n+1)*G(2,2; x)*Z(2,2; n,y), n>=0, where y=c(4*x) and c(x) is the g.f. of A000108 (Catalan) and G(2,2; x) is the g.f. of C(2,2; n+1), that is G(2,2; x)= (c(2,2; x)-1)/x. Hence G(2,2; x)*Z(2,2; k,c(4*x)) is, for k=0,1,..., the g.f. for C(2,2; n+k), n>=0.

Column sequences are: A064340(n), 3*A064340(n+1), Main diagonal gives A002001(n). Row sums give C(2,2; n+1)= A064340(n+1).

Crossrefs

Cf. A067328 (scaled triangle with 1's in main diagonal).

Formula

a(n, 0)= C(2, 2; n) := A064340(n), n>=0; a(n, 1)= 3*C(2, 2; n), n>=1; a(n, m)=4*sum(a(n-1, k), k=(m-1)..(n-1)) if n>=m>=2, else 0.