A318003 E.g.f. A(x) satisfies: A(x) = sinh(x) + cosh(x)*A(x)^2.

1, 2, 13, 140, 2041, 38222, 874693, 23644280, 737301361, 26053773242, 1028890245373, 44906842244420, 2146597351615081, 111529829156824262, 6258181131400784053, 377167403797348584560, 24298520283720455935201, 1666382133585488471159282, 121205126078549481910218733, 9319638200814732292237048700

Offset: 1

Author

Paul D. Hanna, Aug 28 2018

Keywords

• nonn

Examples

			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.
		

Crossrefs

Cf. A318002, A318007.

Programs

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), ", "))

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