A195947 - OEIS

%I #13 Mar 30 2012 18:37:29

%S 1,1,5,58,1093,28731,971719,40236449,1972617385,111779567596,

%T 7189852342091,517600784497237,41237095369088029,3602389000897583001,

%U 342422738142493542031,35186740743134660359186,3887047020291801938191057,459397561144034558519708403

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

%e E.g.f.: A(x) = 1 + x + 5*x^2/2! + 58*x^3/3! + 1093*x^4/4! + 28731*x^5/5! +...

%e where:

%e A(x) = 1 + A(x)*exp(A(x) - 1)*x + A(x)^4*exp(A(x)^2 - 1)*x^2/2! + A(x)^9*exp(A(x)^3 - 1)*x^3/3! + A(x)^16*exp(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. A196022, A195895.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 27 2011