OFFSET
0,3
FORMULA
Given g.f. A(x), define G(x) by G(x)/x = g.f. of A088717, then G(x) satisfies:
(1) G(x) = x + x*G(G(x)^2/x),
(2) G(x) = Series_Reversion(x/A(x)),
(3) G(x) = x/Series_Reversion(x*A(x)) - 1,
(4) G(x) = x*A(G(x)),
(5) G(x) = A(x/(1+G(x))) - 1,
(6) A(x) = 1 + G(x*A(x)).
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 35*x^4 + 220*x^5 + 1622*x^6 +...
such that A(x) = 1 + x*A(x)*A(A(x)-1) where:
A(A(x)-1) = 1 + x + 4*x^2 + 22*x^3 + 148*x^4 + 1147*x^5 + 9901*x^6 +...
Let G(x) satisfy G(x/A(x)) = x, then G(x)/x = g.f. of A088717, where
G(x) = x + x^2 + 3*x^3 + 14*x^4 + 84*x^5 + 596*x^6 + 4785*x^7 +...
A(G(x)) = 1 + x + 3*x^2 + 14*x^3 + 84*x^4 + 596*x^5 + 4785*x^6 +...
A(x/(1+G(x))) = 1 + x + x^2 + 3*x^3 + 14*x^4 + 84*x^5 + 596*x^6 + 4785*x^7 +...
G(x*A(x)) = x + 2*x^2 + 7*x^3 + 35*x^4 + 220*x^5 + 1622*x^6 +...
PROG
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+x*A*subst(A, x, A-1)+x*O(x^n)); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+subst(serreverse(x/A), x, x*A)+x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 19 2012
STATUS
approved