OFFSET
0,3
FORMULA
E.g.f. satisfies: A(x) = exp( Series_Reversion( Integral 1/A(x)^2 dx ) ).
EXAMPLE
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 201*x^4/4! + 3097*x^5/5! + 63963*x^6/6! +...
Related expansions.
A(x)^2 = 1 + 2*x + 8*x^2/2! + 56*x^3/3! + 608*x^4/4! + 9344*x^5/5! + 190400*x^6/6! +...+ A233335(n)*(2*x)^n/n! +...
Integral 1/A(x)^2 dx = x - 2*x^2/2! - 8*x^4/4! - 96*x^5/5! - 1664*x^6/6! +...
The series reversion of the Integral 1/A(x)^2 dx equals log(A(x)) and begins:
log(A(x)) = x + 2*x^2/2! + 12*x^3/3! + 128*x^4/4! + 2016*x^5/5! + 42656*x^6/6! + 1145280*x^7/7! + 37563008*x^8/8! +...+ A259267(n)*x^n/n! +...
PROG
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(serreverse(intformal(1/A^2+x*O(x^n))))); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 07 2013
STATUS
approved