OFFSET
1,2
COMMENTS
Compare to: C(x) = x*(1 + 2*C(x)*C'(x)) / (1 + C(x)*C'(x)) holds when C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..400
FORMULA
a(n) ~ c * 2^n * n!, where c = 0.181799839377767875340143846... - Vaclav Kotesovec, Oct 14 2020
EXAMPLE
G.f.: A(x) = x + 2*x^2 + 10*x^3 + 74*x^4 + 710*x^5 + 8322*x^6 + 115018*x^7 + 1828962*x^8 + 32852526*x^9 + 657188258*x^10 + ...
such that A = A(x) satisfies: A = x*(1 + 3*A*A')/(1 + A*A').
RELATED SERIES.
A(x)*A'(x) = x + 6*x^2 + 48*x^3 + 470*x^4 + 5448*x^5 + 73374*x^6 + 1132000*x^7 + 19752822*x^8 + 385285080*x^9 + 8311631702*x^10 + ...
PROG
(PARI) {a(n) = my(L=x); for(i=1, n, L = x*(1 + 3*L'*L)/(1 + L'*L +x*O(x^n)) ); polcoeff(L, n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 28 2018
STATUS
approved