OFFSET
0,3
FORMULA
E.g.f. satisfies: A(x) = Sum_{n>=0} A(x)^(-n^2) * exp(1/A(x)^n - 1)*x^n/n!.
EXAMPLE
E.g.f.: A(x) = 1 + x - 3*x^2/2! + 22*x^3/3! - 227*x^4/4! + 2571*x^5/5! +...
where:
A(x) = 1 + 1/A(x)*exp(1/A(x) - 1)*x + 1/A(x)^4*exp(1/A(x)^2 - 1)*x^2/2! + 1/A(x)^9*exp(1/A(x)^3 - 1)*x^3/3! + 1/A(x)^16*exp(1/A(x)^4 - 1)*x^4/4! +...
Also, e.g.f. A = A(x) satisfies:
A(x) = 1 - (1 - (1+x/A)) + 1/2!*(1 - 2*(1+x/A) + (1+x/A^2)^2) -
1/3!*(1 - 3*(1+x/A) + 3*(1+x/A^2)^2 - (1+x/A^3)^3) +
1/4!*(1 - 4*(1+x/A) + 6*(1+x/A^2)^2 - 4*(1+x/A^3)^3 + (1+x/A^4)^4) -
1/5!*(1 - 5*(1+x/A) + 10*(1+x/A^2)^2 - 10*(1+x/A^3)^3 + 5*(1+x/A^4)^4 - (1+x/A^5)^5) +-...
PROG
(PARI) {a(n)=local(A=1+x, X=x+x*O(x^n)); for(i=1, n, A=1+sum(m=1, n, exp(A^(-m)-1)*A^(-m^2)*X^m/m!)); n!*polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x, X=x+x*O(x^n)); for(i=1, n, A=1+sum(m=1, n, 1/m!*sum(k=0, m, binomial(m, k)*(-1)^(m-k)*(1+X*A^(-k))^k))); n!*polcoeff(A, n)}
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Oct 08 2011
STATUS
approved