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A143599 E.g.f. satisfies: A(x) = exp( x*sqrt(A(x)/A(-x)) ). 2

%I

%S 1,1,3,10,53,316,2527,22072,239689,2774800,38284091,553477024,

%T 9284250109,161180444608,3187413648343,64638167906176,

%U 1473221217774353,34190645940363520,882759869810501491,23079229227696318976

%N E.g.f. satisfies: A(x) = exp( x*sqrt(A(x)/A(-x)) ).

%H Vaclav Kotesovec, <a href="/A143599/b143599.txt">Table of n, a(n) for n = 0..218</a>

%F E.g.f.: A(x) = exp(x*exp(x*G(x))) where G(x) = cosh(x*G(x)) = e.g.f. of A143601.

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

%F E.g.f.: [sqrt(A(x)/A(-x)) + sqrt(A(-x)/A(x))]/2 = e.g.f. of A143601.

%F E.g.f.: [sqrt(A(x)/A(-x)) - sqrt(A(-x)/A(x))]/2 = e.g.f. of A007106.

%F E.g.f.: A(x) = H(x/2)^2 where H(x) = exp(x*H(x)/H(-x)) = e.g.f. of A143600.

%F E.g.f. satisfies: A(x/cosh(x)) = exp(x*exp(x)/cosh(x)). [From _Paul D. Hanna_, Aug 29 2008]

%F E.g.f. satisfies: -x*sqrt(x^2/log(y)^2) = log((x^2*y)/log(y)^2), where y=A(x). - _Vaclav Kotesovec_, Feb 26 2014

%F a(n) ~ c * n! * d^n / n^(3/2), where d = 1.5088795615383199289... is the root of the equation sqrt(1+1/d^2) = 1 + LambertW((1+sqrt(1+1/d^2))/exp(1 + sqrt(1+1/d^2))), and c = 7.98255033020099890281693169... if n is even, and c = 7.852067808737280621088934789... if n is odd. - _Vaclav Kotesovec_, Feb 26 2014

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 10*x^3/3! + 53*x^4/4! + 316*x^5/5! +...

%e F(x) = sqrt(A(x)/A(-x)) = e.g.f. of A058014:

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

%e where F(x) = exp(x*(F(x) + 1/F(x))/2).

%e G(x) = [sqrt(A(x)/A(-x)) + sqrt(A(-x)/A(x))]/2 = e.g.f. of A143601:

%e G(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! +...

%e where G(x) = cosh(x*G(x)).

%e S(x) = [sqrt(A(x)/A(-x)) - sqrt(A(-x)/A(x))]/2 = e.g.f. of A007106:

%e S(x) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + 686080*x^9/9! +...

%e where S(x) = sqrt(G(x)^2 - 1) and G(x) = e.g.f. of A143601.

%o (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=exp(x*sqrt(A/subst(A,x,-x))));n!*polcoeff(A,n)}

%Y Cf. A058014, A143600, A143601, A007106.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Aug 27 2008

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Last modified August 20 01:14 EDT 2019. Contains 326136 sequences. (Running on oeis4.)