A196959 - OEIS

%I #5 Mar 30 2012 18:37:31

%S 1,1,9,193,6721,326001,20316937,1548374129,139576777921,

%T 14530808439073,1715928199384521,226652340142349793,

%U 33113449456084235905,5302086923264289694225,923349950199153833740105,173761214485224395469845521,35139709415689684107278235265

%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)^(2*k))^k.

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

%e  E.g.f.: A(x) = 1 + x + 9*x^2/2! + 193*x^3/3! + 6721*x^4/4! + 326001*x^5/5! +...

%e where:

%e A(x) = 1 + A(x)^2*exp(A(x)^2 - 1)*x + A(x)^8*exp(A(x)^4 - 1)*x^2/2! + A(x)^18*exp(A(x)^6 - 1)*x^3/3! + A(x)^32*exp(A(x)^8 - 1)*x^4/4! +...

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

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

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

%e 1/4!*(1 - 4*(1+x*A^2) + 6*(1+x*A^4)^2 - 4*(1+x*A^6)^3 + (1+x*A^8)^4) -

%e 1/5!*(1 - 5*(1+x*A^2) + 10*(1+x*A^4)^2 - 10*(1+x*A^6)^3 + 5*(1+x*A^8)^4 - (1+x*A^10)^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^(2*m)-1)*A^(2*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^(2*k))^k))); n!*polcoeff(A, n)}

%Y  Cf. A195947, A196958.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 08 2011