0,3

E.g.f. satisfies: A(x) = cosh(x*A(x)).

E.g.f.: A(x) = (1/x)*Series_Reversion( x/cosh(x) ).

E.g.f.: sqrt(A(x)^2 - 1) = e.g.f. of A007106.

E.g.f.: exp(x*A(x)) = A(x) + sqrt(A(x)^2-1) = e.g.f. of A058014.

E.g.f.: A(x) = [F(x) + F(-x)]/2 where F(x) = exp(x*[F(x) + 1/F(x)]/2) = e.g.f. of A058014.

E.g.f.: A(2x) = [G(x)/G(-x) + G(-x)/G(x)]/2 where G(x) = exp(x*G(x)/G(-x)) = e.g.f. of A143600.

Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Aug 29 2008: (Start)

E.g.f. satisfies: A(x/cosh(x)) = cosh(x).

a(n) = (2n)!*[x^(2n)] cosh(x)^(2n+1)/(2n+1). (End)

E.g.f.: A(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! +...

The e.g.f. of A007106 (a bisection of A058014) is given by:

sqrt(A(x)^2 - 1) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + 686080*x^9/9! +...

The e.g.f. of A058014 is given by:

F(x) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 96*x^5/5! + 541*x^6/6! +...

where A(x) = [F(x) + F(-x)]/2 and exp(x*A(x)) = F(x).

The e.g.f. of A143600 is given by:

G(x) = 1 + x + 5*x^2/2! + 25*x^3/3! + 249*x^4/4! + 2561*x^5/5! +...

where A(2x) = [G(x)/G(-x) + G(-x)/G(x)]/2.

(PARI) {a(n)=local(A=1+x*O(x^n)); for(i=0, n, A=cosh(x*A)); n!*polcoeff(A, n)}

(PARI) {a(n)=(2*n)!*polcoeff(cosh(x+x*O(x^(2*n)))^(2*n+1)/(2*n+1), 2*n)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Aug 29 2008]

Cf. A058014, A143600, A007106.

Sequence in context: A139168 A030256 A023332 this_sequence A050286 A096761 A076461

Adjacent sequences: A143598 A143599 A143600 this_sequence A143602 A143603 A143604

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Aug 26 2008, May 27 2009

Edited by Paul D. Hanna (pauldhanna(AT)juno.com), May 27 2009