A195511 - OEIS

%I #11 Jan 12 2014 11:16:54

%S 1,1,3,19,197,2801,50407,1098371,28122761,827684785,27534518411,

%T 1021777860995,41847737874637,1875044409274817,91239372967844207,

%U 4791502346638758931,270114113377777911569,16269795487513345957025,1042794341136010753491475

%N E.g.f. satisfies: A(x) = (exp(x) + exp(x*A(x)^2))/2.

%H Vaclav Kotesovec, <a href="/A195511/b195511.txt">Table of n, a(n) for n = 0..100</a>

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

%F E.g.f. satisfies: A(x) = C(x*A(x)) where C(x) = A(x/C(x)) is the g.f. of A195510 and satisfies: C(x) = (exp(x*C(x)) + exp(x/C(x)))/2.

%F a(n) ~ sqrt(((1+2*r)*s^2-1)/(2+4*r*s^2)) * n^(n-1) / (exp(n) * r^n), where r = 0.258248183317928786953777... and s = 1.7522591181936492232545... are the roots of the equations exp(r) + exp(r*s^2) = 2*s, exp(r*s^2)*r*s = 1. - _Vaclav Kotesovec_, Jan 11 2014

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 197*x^4/4! + 2801*x^5/5! +...

%e Related series begin:

%e exp(x*A(x)^2) = 1 + x + 5*x^2/2! + 37*x^3/3! + 393*x^4/4! + 5601*x^5/5! +...

%e A(x)^2 = 1 + 2*x + 8*x^2/2! + 56*x^3/3! + 600*x^4/4! + 8712*x^5/5! +...

%e The g.f. C(x) of A195510 begins:

%e C(x) = 1 + x + x^2/2! + 4*x^3/3! + 25*x^4/4! + 156*x^5/5! + 1561*x^6/6! +...

%e where A(x/C(x)) = C(x) = (exp(x*C(x)) + exp(x/C(x)))/2.

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

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 19 2011