0,4

David Callan, A Combinatorial Interpretation of the Eigensequence for Composition, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.4.

N. J. A. Sloane, Transforms

G.f. A(x) satisfies the functional equation: A(x)-x = x*A(A(x)). - Paul D. Hanna (pauldhanna(AT)juno.com), Aug 04 2002

G.f.: A(x/(1+A(x))) = x. - Paul D. Hanna (pauldhanna(AT)juno.com), Dec 04 2003

Suppose functions A=A(x), B=B(x), C=C(x), etc., satisfy: A = 1 + xAB, B = 1 + xABC, C = 1 + xABCD, D = 1 + xABCDE, etc., then B(x)=A(x*A(x)), C(x)=B(x*A(x)), D(x)=C(x*A(x)), etc., where A(x) = 1 + x*A(x)*A(x*A(x)) and x*A(x) is the g.f. of this sequence (see table A128325). - Paul D. Hanna (pauldhanna(AT)juno.com), Mar 10 2007

G.f. A(x) = x*F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1)*(1 + x*F(x,n+1)) for n>0 with F(x,0)=1. - Paul D. Hanna (pauldhanna(AT)juno.com), Apr 16 2007

a(n) = [x^(n-1)] [1 + A(x)]^n/n for n>=1 with a(0)=0; i.e., a(n) equals the coefficient of x^(n-1) in [1+A(x)]^n/n for n>=1. [From Paul D. Hanna (pauldhanna(AT)juno.com), Nov 18 2008]

Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Jul 09 2009: (Start)

Let A(x)^m = Sum_{n>=0} a(n,m)*x^n with a(0,m)=1, then

a(n,m) = Sum_{k=0..n} m*C(n+m,k)/(n+m) * a(n-k,k).

(End)

(PARI) {a(n)=local(A=1+x); for(i=0, n, A=1+x*A*subst(A, x, x*A+x*O(x^n))); polcoeff(A, n)} - Paul D. Hanna (pauldhanna(AT)juno.com), Mar 10 2007

(PARI) {a(n)=local(A=sum(i=1, n-1, a(i)*x^i)+x*O(x^n)); if(n==0, 0, polcoeff((1+A)^n/n, n-1))} [From Paul D. Hanna (pauldhanna(AT)juno.com), Nov 18 2008]

(PARI) {a(n, m=1)=if(n==0, 1, if(m==0, 0^n, sum(k=0, n, m*binomial(n+m, k)/(n+m)*a(n-k, k))))} [From Paul D. Hanna (pauldhanna(AT)juno.com), Jul 09 2009]

Cf. A001028, A035049.

Cf. A110447.

Cf. A128325.

Sequence in context: A117106 A137534 A137535 this_sequence A110447 A137536 A137537

Adjacent sequences: A030263 A030264 A030265 this_sequence A030267 A030268 A030269

nonn,nice,eigen

Christian G. Bower (bowerc(AT)usa.net)