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

0,3

Comments

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

Links

Table of n, a(n) for n=0..30.

Example

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.

Prog

(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)}

Crossrefs

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

Sequence in context: A145267 A103287 A136704 * A101785 A363306 A003089

Adjacent sequences: A120892 A120893 A120894 * A120896 A120897 A120898

Keyword

nonn

Author

Paul D. Hanna, Jul 14 2006

Status

approved