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E.g.f. satisfies: A(x) = x/(1 - sinh(A(x))).
4

%I #8 Jan 12 2014 11:16:14

%S 1,2,12,124,1800,33606,766976,20689208,643996800,22719618250,

%T 895853071872,39043448067636,1863697724715008,96698693656306574,

%U 5418685033626992640,326140667283301420912,20983722785088536346624,1437191703493403790787218,104400577820040681757736960

%N E.g.f. satisfies: A(x) = x/(1 - sinh(A(x))).

%F E.g.f. A(x) satisfies:

%F (1) A(x - x*sinh(x)) = x.

%F (2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*sinh(x)^n/n!.

%F (3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*sinh(x)^n/n! ).

%F a(n) = n*A201628(n-1).

%F a(n) = (n-1)! * [x^n] x/(1 - sinh(x))^n.

%F a(n) ~ n^(n-1) / (sqrt(s+(2-s^2)*cosh(s)) * exp(n) * (s^2*cosh(s))^(n-1/2)), where s = 0.465767175470891411756875... is the root of the equation s*cosh(s) = 1-sinh(s). - _Vaclav Kotesovec_, Jan 12 2014

%e E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 124*x^4/4! + 1800*x^5/5! +...

%e Related expansions:

%e A(x) = x + x*sinh(x) + d/dx x^2*sinh(x)^2/2! + d^2/dx^2 x^3*sinh(x)^3/3! + d^3/dx^3 x^4*sinh(x)^4/4! +...

%e log(A(x)/x) = sinh(x) + d/dx x*sinh(x)^2/2! + d^2/dx^2 x^2*sinh(x)^3/3! + d^3/dx^3 x^3*sinh(x)^4/4! +...

%e A(x)/x = 1 + x + 4*x^2/2! + 31*x^3/3! + 360*x^4/4! + 5601*x^5/5! + 109568*x^6/6! +...+ A201628(n)*x^n/n! +...

%e sinh(A(x)) = x + 2*x^2/2! + 13*x^3/3! + 136*x^4/4! + 1981*x^5/5! + 37056*x^6/6! + 846777*x^7/7! + 22861952*x^8/8! +...

%t Rest[CoefficientList[InverseSeries[Series[x - x*Sinh[x],{x,0,20}],x],x] * Range[0,20]!] (* _Vaclav Kotesovec_, Jan 12 2014 *)

%o (PARI) {a(n)=(n-1)!*polcoeff(x/(1 - sinh(x+x*O(x^n)))^n,n)}

%o for(n=1, 25, print1(a(n), ", "))

%o (PARI) {a(n)=n!*polcoeff(serreverse(x-x*sinh(x+x*O(x^n))), n)}

%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^m*sinh(x+x*O(x^n))^m/m!)); n!*polcoeff(A, n)}

%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*sinh(x+x*O(x^n))^m/m!)+x*O(x^n))); n!*polcoeff(A, n)}

%Y Cf. A201628, A214222, A214224, A214225.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jul 07 2012