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E.g.f. satisfies: A(x) = Sum_{n>=0} 1/n! * Sum_{k=0..n} (-1)^(n-k) * C(n,k) * (1 + x/A(x)^k)^k.
1

%I #6 Mar 30 2012 18:37:31

%S 1,1,-3,22,-227,2571,-19157,-550675,47287609,-2474401796,113036728791,

%T -4672627704315,162246902824213,-2986895872839215,-218043087879704765,

%U 36487218926663045686,-3474880515053581779215,262843589524537015935667,-15730145172651453469201745,541394288749029235105442821

%N E.g.f. satisfies: A(x) = Sum_{n>=0} 1/n! * Sum_{k=0..n} (-1)^(n-k) * C(n,k) * (1 + x/A(x)^k)^k.

%F E.g.f. satisfies: A(x) = Sum_{n>=0} A(x)^(-n^2) * exp(1/A(x)^n - 1)*x^n/n!.

%e E.g.f.: A(x) = 1 + x - 3*x^2/2! + 22*x^3/3! - 227*x^4/4! + 2571*x^5/5! +...

%e where:

%e 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! +...

%e Also, e.g.f. A = A(x) satisfies:

%e A(x) = 1 - (1 - (1+x/A)) + 1/2!*(1 - 2*(1+x/A) + (1+x/A^2)^2) -

%e 1/3!*(1 - 3*(1+x/A) + 3*(1+x/A^2)^2 - (1+x/A^3)^3) +

%e 1/4!*(1 - 4*(1+x/A) + 6*(1+x/A^2)^2 - 4*(1+x/A^3)^3 + (1+x/A^4)^4) -

%e 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) +-...

%o (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)}

%o (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)}

%Y Cf. A195947, A196959.

%K sign

%O 0,3

%A _Paul D. Hanna_, Oct 08 2011