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%I #7 Sep 06 2018 07:03:10
%S 1,2,13,140,2041,38222,874693,23644280,737301361,26053773242,
%T 1028890245373,44906842244420,2146597351615081,111529829156824262,
%U 6258181131400784053,377167403797348584560,24298520283720455935201,1666382133585488471159282,121205126078549481910218733,9319638200814732292237048700
%N E.g.f. A(x) satisfies: A(x) = sinh(x) + cosh(x)*A(x)^2.
%F E.g.f. A(x) satisfies:
%F (1) A(x) = sinh(x) + cosh(x)*A(x)^2.
%F (2) A(x) = sinh(x) * Sum_{n>=0} binomial(2*n,n)/(n+1) * sinh(2*x)^n/2^n.
%F (3) A(x) = (1 - sqrt(1 - 2*sinh(2*x))) / (2*cosh(x)).
%F (4) A(x) = 2*sinh(x) / (1 + sqrt(1 - 2*sinh(2*x))).
%F 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
%e 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! + ...
%e such that A(x) = sinh(x) + cosh(x)*A(x)^2.
%o (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)}
%o for(n=1, 25, print1(a(n), ", "))
%Y Cf. A318002, A318007.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Aug 28 2018
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