OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..300
FORMULA
G.f. satisfies: A(x) = (1 + 2*x*A(x)^2 - x^2*A(x)^3) / (1+x - 2*x^2*A'(x)).
a(n) ~ c * n * 2^n * n!, where c = 0.1840416364326449945692... - Vaclav Kotesovec, Aug 24 2017
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 29*x^3 + 286*x^4 + 3478*x^5 + 49750*x^6 +...
such that
log(1+x + x^2*A(x)^2) = x + x^2/2 + 4*x^3/3 + 29*x^4/4 + 286*x^5/5 + 3478*x^6/6 + 49750*x^7/7 +...+ a(n-1)*x^n/n +...
Related expansions.
1+x + x^2*A(x)^2 = 1 + x + x^2 + 2*x^3 + 9*x^4 + 66*x^5 + 646*x^6 + 7760*x^7 109585*x^8 +...+ A259607(n)*x^n +...
A(x)^2 = 1 + 2*x + 9*x^2 + 66*x^3 + 646*x^4 + 7760*x^5 + 109585*x^6 +...+ A259607(n+2)*x^n +...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=deriv(log(1+x + x^2*A^2 +x^2*O(x^n)))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 26 2012
STATUS
approved