OFFSET
1,2
FORMULA
E.g.f. A(x) satisfies:
(1) A(x) = sinh(x) + cosh(x)*A(x)^2.
(2) A(x) = sinh(x) * Sum_{n>=0} binomial(2*n,n)/(n+1) * sinh(2*x)^n/2^n.
(3) A(x) = (1 - sqrt(1 - 2*sinh(2*x))) / (2*cosh(x)).
(4) A(x) = 2*sinh(x) / (1 + sqrt(1 - 2*sinh(2*x))).
a(n) ~ 5^(1/4) * sinh(log(phi)/2) * 2^(n + 1/2) * n^(n-1) / (log(phi)^(n - 1/2) * exp(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 06 2018
EXAMPLE
E.g.f.: A(x) = x + 2*x^2/2! + 13*x^3/3! + 140*x^4/4! + 2041*x^5/5! + 38222*x^6/6! + 874693*x^7/7! + 23644280*x^8/8! + 737301361*x^9/9! + 26053773242*x^10/10! + ...
such that A(x) = sinh(x) + cosh(x)*A(x)^2.
PROG
(PARI) {a(n) = my(A = 2*sinh(x +x^2*O(x^n)) / (1 + sqrt(1 - 2*sinh(2*x +x^2*O(x^n)))) ); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 28 2018
STATUS
approved