OFFSET
0,3
COMMENTS
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..250
FORMULA
G.f. satisfies: 1+x + x*A'(x)/A(x) = d/dx x^2/Series_Reversion(x*A(x)).
a(n) ~ c * (n!)^2 / sqrt(n), where c = 0.500612869985729164508780668394780439... - Vaclav Kotesovec, Oct 18 2017
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 140*x^4 + 3102*x^5 +...
Compare the series S(x) = d/dx x^2/Series_Reversion(x*A(x)):
S(x) = 1 + 2*x + 3*x^2 + 28*x^3 + 515*x^4 + 14766*x^5 + 596652*x^6 +...
to the logarithmic derivative:
A'(x)/A(x) = 1 + 3*x + 28*x^2 + 515*x^3 + 14766*x^4 + 596652*x^5 +...
and also to the g.f. G(x) of A177752:
G(x) = 1 + x + x^2 + 7*x^3 + 103*x^4 + 2461*x^5 + 85236*x^6 +...
PROG
(PARI) {a(n)=local(A=1+x+sum(m=2, n-1, a(m)*x^m)); A=(1/x)*serreverse(x^2/intformal(1+x+x*deriv(A)/(A+x*O(x^n)))); if(n<0, 0, if(n<2, 1, polcoeff((n+1)*A, n)))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 16 2010
STATUS
approved