0,3

Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Nov 15 2008: (Start)

More generally, if g.f. A(x) satisfies: A(x/A(x)^k) = 1 + x*A(x)^m, then

A(x) = 1 + x*G(x)^(m+k) where G(x) = A(x*G(x)^k) and G(x/A(x)^k) = A(x);

thus a(n) = [x^(n-1)] ((m+k)/(m+k*n))*A(x)^(m+k*n) for n>=1 with a(0)=1. (End)

G.f. satisfies: A(x) = 1 + x*G(x)^2 where G(x) = g.f. of A121687.

G.f. satisfies: A(x) = G(x/A(x)) where G(x) = A(x*G(x)) = g.f. of A121687. [From Paul D. Hanna (pauldhanna(AT)juno.com), Nov 08 2008]

a(n) = [x^(n-1)] (2/(n+1))*A(x)^(n+1) for n>=1 with a(0)=1; i.e., a(n) equals 2/(n+1) times the coefficient of x^(n-1) in A(x)^(n+1) for n>=1. [From Paul D. Hanna (pauldhanna(AT)juno.com), Nov 15 2008]

G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 34*x^4 + 203*x^5 + 1398*x^6 +..

A(x/A(x)) = 1 + x + x^2 + 2*x^3 + 7*x^4 + 34*x^5 + 203*x^6 + 1398*x^7 +..

A(x) = 1 + x*G(x)^2 where

G(x) = 1 + x + 3*x^2 + 14*x^3 + 83*x^4 + 574*x^5 + 4432*x^6 +...

is the g.f. of A121687.

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

(PARI) {a(n)=local(F=1+x); for(i=0, n, G=serreverse(x/(F+x*O(x^n)))/x; F=1+x*G^2); polcoeff(F, n)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Nov 08 2008]

(PARI) /* This sequence is generated when k=1, m=1: A(x/A(x)^k) = 1 + x*A(x)^m */ {a(n, k=1, m=1)=local(A=sum(i=0, n-1, a(i, k, m)*x^i)); if(n==0, 1, polcoeff((m+k)/(m+k*n)*A^(m+k*n), n-1))} [From Paul D. Hanna (pauldhanna(AT)juno.com), Nov 15 2008]

Cf. A121687.

Sequence in context: A058915 A020054 A117399 this_sequence A056543 A075834 A011800

Adjacent sequences: A145342 A145343 A145344 this_sequence A145346 A145347 A145348

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Nov 05 2008