OFFSET
1,2
COMMENTS
First negative term is a(27).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..100
FORMULA
E.g.f. A(x) satisfies:
(1) A(-A(-x)) = x.
(2a) 1 = Sum_{n>=0} (-1)^n * ( A(x) - (-1)^n*A(-x) )^n/n!.
(2b) 1 = Sum_{n>=0} ( x + (-1)^n*A(A(x)) )^n/n!.
(3a) 1 = cosh(A(x) - A(-x)) - sinh(A(x) + A(-x)).
(3b) 1 = cosh(A(-x))*exp(-A(x)) - sinh(A(-x))*exp(A(x)).
(3c) 1 = cosh(x)*exp(-A(A(x))) + sinh(x)*exp(A(A(x))).
(4a) A(x) = log( 2*cosh(A(-x)) / (1 + sqrt(1 + 2*sinh(2*A(-x)))) ).
(4b) A(x) = log( (sqrt(1 + 2*sinh(2*A(-x))) - 1) / (2*sinh(A(-x))) ).
(5) A(A(x)) = log( 2*cosh(x) / (1 + sqrt(1 - 2*sinh(2*x))) ), which is the e.g.f. of A318000.
EXAMPLE
E.g.f.: A(x) = x + 2*x^2/2! + 6*x^3/3! + 56*x^4/4! + 600*x^5/5! + 8432*x^6/6! + 144816*x^7/7! + 2892416*x^8/8! + 66721920*x^9/9! + 1732489472*x^10/10! + 50144683776*x^11/11! + 1604936139776*x^12/12! + 56236356234240*x^13/13! + 2137961925773312*x^14/14! + 87642967518836736*x^15/15! + ...
such that cosh(A(x) - A(-x)) - sinh(A(x) + A(-x)) = 1.
RELATED SERIES.
(1) exp(A(x)) = 1 + x + 3*x^2/2! + 13*x^3/3! + 105*x^4/4! + 1141*x^5/5! + 16083*x^6/6! + 276193*x^7/7! + 5561265*x^8/8! + 128834761*x^9/9! + 3365571363*x^10/10! + ...
which equals (sqrt(1 + 2*sinh(2*A(-x))) - 1) / (2*sinh(A(-x))).
(2) A(A(x)) = x + 4*x^2/2! + 24*x^3/3! + 256*x^4/4! + 3840*x^5/5! + 73024*x^6/6! + 1688064*x^7/7! + 45991936*x^8/8! + 1443102720*x^9/9! + ... + A318000(n)*x^n/n! + ...
which equals log( 2*cosh(x) / (1 + sqrt(1 - 2*sinh(2*x))) ).
PROG
(PARI) {a(n) = my(A=x+x^2 +x*O(x^n), S=x); for(i=1, n, S = (A - subst(A, x, -x))/2;
A = S + log(cosh(2*S) - 1 + sqrt(1 + (cosh(2*S) - 1)^2))/2;
A = (A - subst(serreverse(A), x, -x))/2 ); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Aug 20 2018
STATUS
approved