_Paul D. Hanna (pauldhanna(AT)juno.com), _, Aug 26 2008

E.g.f. satisfies: A(x) = exp( A(x)*Series_Reversion[x*A(x)] ).

1, 1, 1, 7, -11, 741, -14129, 521263, -20968359, 1063764649, -63316356389, 4408796480331, -352958649497387, 32158017135672013, -3302679619545572265, 379346145007147112551, -48397471256028983134799, 6817654800019973404119633, -1054828080584161260522077645

0,4

E.g.f. satisfies: A(x*A(x)) = exp(x*A(x*A(x))) = LambertW(-x)/(-x).

A(x) = 1 + x + x^2/2! + 7*x^3/3! - 11*x^4/4! + 741*x^5/5! - 14129*x^6/6! +-...

A(x*A(x)) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! +...

LambertW(-x)/(-x) = 1 + x + 3^1*x^2/2! + 4^2*x^3/3! + 5^3*x^4/4! +...

log(A(x)) = x + 2*x^3/2! - 9*x^4/3! + 172*x^5/4! - 3205*x^6/5! +-...

Series_Reversion[x*A(x)] = x - x^2 + 3*x^3/2! - 22*x^4/3! + 281*x^5/4! - 5396*x^6/5! +-...

(PARI) {a(n)=local(A=1); for(i=0, n, A=exp(A*serreverse(x*A+x^2*O(x^n)))); n!*polcoeff(A, n)}

Cf. A000272.

sign

Paul D. Hanna (pauldhanna(AT)juno.com), Aug 26 2008

approved