A195510 E.g.f. satisfies: A(x) = (exp(x*A(x)) + exp(x/A(x)))/2.

1, 1, 1, 4, 25, 156, 1561, 15968, 205521, 2996416, 48245521, 891169632, 17701061929, 390008578688, 9273243108585, 237666643184896, 6557636429111329, 192526245072832512, 6029011794295629217, 199832740470058237952, 7000664928137597341881

Offset: 0

Author

Paul D. Hanna, Sep 19 2011

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 25*x^4/4! + 156*x^5/5! +...
Related series begin:
exp(x*A(x)) = 1 + x + 3*x^2/2! + 10*x^3/3! + 53*x^4/4! + 376*x^5/5! + 3247*x^6/6! +...
exp(x/A(x)) = 1 + x - x^2/2! - 2*x^3/3! - 3*x^4/4! - 64*x^5/5! - 125*x^6/6! -...
The g.f. B(x) of A195511 begins:
B(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 197*x^4/4! + 2801*x^5/5! + 50407*x^6/6! +...
where A(x*B(x)) = B(x) = (exp(x) + exp(x*B(x)^2))/2.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A195511.

Programs

PARI

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

Formula

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

E.g.f. satisfies: A(x) = B(x/A(x)) where B(x) = A(x*B(x)) is the g.f. of A195511 and satisfies: B(x) = (exp(x) + exp(x*B(x)^2))/2.

a(n) ~ sqrt(s^3*(-1+2*r*s+s^2)/(-2*s + r^2*s*(1+s^2) + r*(-1+2*s^2+s^4))) * n^(n-1) / (exp(n) * r^n), where r = 0.53064794557035571... and s = 2.58970226260372420623... are the roots of the equations exp(r/s) + exp(r*s) = 2*s, exp(r*s)*r*s = exp(r/s)*r/s + 2*s. - Vaclav Kotesovec, Jan 11 2014