A119013 Eigenvector of triangle A118588; E.g.f. satisfies: A(x) = exp(x)*A(x^2+x^3).

1, 1, 3, 13, 73, 621, 5491, 60313, 743793, 10115353, 158914531, 2815311621, 55094081593, 1142894689093, 25142695616403, 594557634923281, 15084112106943841, 407999468524242993, 11669035487641120963

Offset: 0

Author

Paul D. Hanna, May 08 2006

Keywords

• nonn

Comments

E.g.f. of triangle A118588 is exp(x + y*(x^2+x^3)); note the similarity to the e.g.f. of this sequence. More generally, the e.g.f. of an eigenvectors can be determined from the e.g.f. of a triangle as follows. [ Given a triangle with e.g.f.: exp(x + y*x*F(x)) such that F(0) = 0, then the eigenvector has e.g.f.: exp(G(x)) where o.g.f. G(x) satisfies: G(x) = x + G(x*F(x)). ]

Examples

			A(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 73*x^4/4! + 621*x^5/5! +...
log(A(x)) = o.g.f. of A014535 =
x + x^2+ x^3+ x^4+ 2*x^5+ 2*x^6+ 3*x^7+ 4*x^8+ 5*x^9+ 8*x^10 +...
		

Crossrefs

Cf. A118588 (triangle), A118589 (row sums), A014535 (log(A(x))).

Programs

PARI

{a(n)=if(n==0,1,sum(k=0,n\2,a(k)*n!*polcoeff(polcoeff(exp(x+y*(x^2+x^3)+x*O(x^n)+y*O(y^k)),n,x),k,y)))}

Formula

Log(A(x)) = o.g.f. of A014535 (B-trees of order 3 with n leaves).