OFFSET
0,3
FORMULA
G.f. A(x), along with F(x) = A(x/F(x)) and G(x) = A(x*G(x)), satisfy:
* A(x/A(x)) = 1 + x*F(x/A(x)) since G(x/A(x)) = A(x);
* A(x*A(x)) = 1 + x*A(x)^2*G(x*A(x)) since F(x*A(x)) = A(x);
* A(x/F(x)^2) = 1 + x*F( x/F(x)^2 )/F(x) since F(x) = G(x/F(x)^2);
* A(x*G(x)^2) = 1 + x*G(x)^3*G( x*G(x)^2 ) since G(x) = F(x*G(x)^2).
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 17*x^4 + 78*x^5 + 423*x^6 +...
The function F(x) = A(x/F(x)) is the g.f. of A184510 and begins:
F(x) = 1 + x + x^2 + x^3 + 4*x^4 + 22*x^5 + 103*x^6 + 565*x^7 + 3650*x^8 +...
The function G(x) = A(x*G(x)) is the g.f. of A184511 and begins:
G(x) = 1 + x + 3*x^2 + 12*x^3 + 58*x^4 + 324*x^5 + 2016*x^6 + 13629*x^7 + 98644*x^8 +...
Related expansions:
A(x*A(x)) = 1 + x + 3*x^2 + 11*x^3 + 47*x^4 + 236*x^5 + 1365*x^6 + 8799*x^7 + 61770*x^8 +...
A(x/A(x)) = 1 + x + x^2 - 2*x^4 - 2*x^5 - 4*x^6 - 55*x^7 - 281*x^8 - 1545*x^9 -...
PROG
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+x*serreverse(x/A)/serreverse(x*A)+x*O(x^n)); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 16 2011
STATUS
approved