OFFSET
1,3
FORMULA
G.f. A(x) satisfies:
(1) A(x) = x + Sum_{n>=1} (1-x)^n * d^(n-1)/dx^(n-1) A(x)^(2*n) / n!.
(2) A(x) = x*exp( Sum_{n>=1} (1-x)^n * d^(n-1)/dx^(n-1) A(x)^(2*n)/x / n! ).
(3) A( (x - A(x)^2) / (1 - A(x)^2) ) = x.
(4) A(A( 1 - 1/((1+x)*(1-A(x)^2)) )) = x.
(5) A(A(x)) = sqrt( (A(x) - x) / (1 - x) ).
EXAMPLE
G.f.: A(x) = x + x^2 + 3*x^3 + 16*x^4 + 108*x^5 + 836*x^6 + 7136*x^7 +...
The g.f. satisfies the series:
A(x) = x + (1-x)*A(x)^2 + (1-x)^2*d/dx A(x)^4/2! + (1-x)^3*d^2/dx^2 A(x)^6/3! + (1-x)^4*d^3/dx^3 A(x)^8/4! +...
as well as the logarithmic series:
log(A(x)/x) = (1-x)*A(x)^2/x + (1-x)^2*[d/dx A(x)^4/x]/2! + (1-x)^3*[d^2/dx^2 A(x)^6/x]/3! + (1-x)^4*[d^3/dx^3 A(x)^8/x]/4! +...
Related expansions:
A(A(x)) = x + 2*x^2 + 8*x^3 + 48*x^4 + 354*x^5 + 2958*x^6 + 27004*x^7 +...
A(A(x))^2 = x^2 + 4*x^3 + 20*x^4 + 128*x^5 + 964*x^6 + 8100*x^7 +...
(A(x)-x)/(1-x) = x^2 + 4*x^3 + 20*x^4 + 128*x^5 + 964*x^6 + 8100*x^7 +...
The series reversion of the g.f. A(x) equals:
(x-A(x)^2)/(1-A(x)^2) = x - x^2 - x^3 - 6*x^4 - 34*x^5 - 234*x^6 - 1818*x^7 -...
The series reversion of A(A(x)) equals:
1 - 1/((1+x)*(1-A(x)^2)) = x - 2*x^2 - 8*x^4 - 34*x^5 - 242*x^6 - 1852*x^7 -...
PROG
(PARI) {a(n)=local(A=x); for(i=1, n, A=x+(1-x)*subst(A, x, A+x*O(x^n))^2); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse((x - A^2)/(1-A^2+x*O(x^n)))); polcoeff(A, n))}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, (1-x)^m*Dx(m-1, A^(2*m))/m!)+x*O(x^n)); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=1, n, (1-x)^m*Dx(m-1, A^(2*m)/x)/m!)+x*O(x^n))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 13 2012
STATUS
approved