A120895 G.f. satisfies: A(x) = G(x)*A(x^3*G(x)^2) where G(x) is the g.f. of the Motzkin numbers (A001006).

1, 1, 2, 5, 12, 30, 78, 206, 552, 1498, 4105, 11340, 31541, 88237, 248076, 700478, 1985397, 5646129, 16104378, 46056513, 132031176, 379315946, 1091890772, 3148736064, 9095091878, 26310816944, 76219704957, 221085782559, 642058752476, 1866693825362, 5432795508417

Offset: 0

Author

Paul D. Hanna, Jul 14 2006

Keywords

• nonn

Comments

Equals column 0 and main diagonal of triangle A120894 (cascadence of 1+x+x^2).

Examples

			A(x) = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 30*x^5 + 78*x^6 + 206*x^7+...
= G(x)*A(x^3*G(x)^2) where
G(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 21*x^5 + 51*x^6 + 127*x^7 +...
is the g.f. of the Motzkin numbers (A001006) so that G(x) satisfies:
G(x) = 1 + x*G(x) + x^2*G(x)^2.
		

Crossrefs

Cf. A120894, A120896, A120897; A001006; variants: A092684, A092687, A120899.

Programs

PARI

{a(n)=local(A=1+x,G=1/x*serreverse(x/(1+x+x^2+x*O(x^n)))); for(i=0,n,A=G*subst(A,x,x^3*G^2 +x*O(x^n)));polcoeff(A,n,x)}