OFFSET
2,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 2..302
FORMULA
G.f. satisfies:
(1) A(x) = ( x + A( sqrt(A(x)) ) )^2.
(2) A(x) = ( x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^n / n! )^2.
(3) A(x) = x^2 * exp( 2*Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^n/x / n! ).
(4) A(x) = x^2 * G( A(x)/x )^2 where G(x) = 1 + x*G( 1 - 1/G(x) )^2 is the g.f. of A212411.
(5) A(x) = x^2 * F( sqrt(A(x)) )^2 where F(x) = 1 - (x^2/F(x)) / F( x^2/F(x) ) is the g.f. of A213628.
EXAMPLE
G.f.: A(x) = x^2 + 2*x^3 + 9*x^4 + 56*x^5 + 420*x^6 + 3572*x^7 + 33328*x^8 + 334354*x^9 + 3559310*x^10 + 39838760*x^11 + 465743720*x^12 +...
such that A( x - A(x) ) = x^2.
PROG
(PARI) {a(n) = local(A=x^2); for(i=1, n, A = serreverse(x - A +x*O(x^n))^2); polcoeff(A, n)}
for(n=2, 30, print1(a(n), ", "))
(PARI) {Dx(n, F) = local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = local(A=x^2 +x*O(x^n)); for(i=1, n, A = (x + sum(m=1, n, Dx(m-1, A^m)/m!) +x*O(x^n))^2); polcoeff(A, n)}
for(n=2, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 09 2016
STATUS
approved