|
%I #11 Jan 13 2014 09:18:40
%S 1,2,12,132,2040,40350,974400,27805736,915505920,34160797530,
%T 1424581678080,65660547312492,3314551571595264,181866769617012662,
%U 10777121944589844480,685937077729538151120,46668919680893409361920,3380042082757952844150066,259638732115410022642483200
%N E.g.f. satisfies: A(x) = x + A(x)^2*cosh(A(x)).
%F E.g.f. satisfies:
%F (1) Series_Reversion(x - x^2*cosh(x)).
%F (2) x + Sum_{n>=1} d^(n-1)/dx^(n-1) cosh(x)^n*x^(2*n) / n!.
%F (3) x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) cosh(x)^n*x^(2*n-1) / n! ).
%F a(n) = [x^n/n!] x/(1 - x*cosh(x))^n / n for n>0.
%F a(n) ~ s*sqrt(r/(6*r-2*s-r*s^2+s^3)) * n^(n-1) / (exp(n) * r^n), where s = 0.4227473416936597149... is the root of the equation s*(2*cosh(s) + s*sinh(s)) = 1, and r = s - s^2*cosh(s) = 0.22782318947143997934... - _Vaclav Kotesovec_, Jan 13 2014
%e E.g.f: A(x) = x + 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2040*x^5/5! +...
%e Series expressions:
%e A(x) = x + cosh(x)*x^2 + d/dx cosh(x)^2*x^4/2! + d^2/dx^2 cosh(x)^3*x^6/3! + d^3/dx^3 cosh(x)^4*x^8/4! +...
%e log(A(x)/x) = cosh(x)*x + d/dx cosh(x)^2*x^3/2! + d^2/dx^2 cosh(x)^3*x^5/3! + d^3/dx^3 cosh(x)^4*x^7/4! +...
%t Rest[CoefficientList[InverseSeries[Series[x - x^2*Cosh[x], {x, 0, 20}], x],x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Jan 13 2014 *)
%o (PARI) {a(n)=n!*polcoeff(serreverse(x-x^2*cosh(x+x*O(x^n))), n)}
%o for(n=1, 25, print1(a(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, cosh(x+x*O(x^n))^m*x^(2*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, cosh(x+x*O(x^n))^m*x^(2*m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
%o (PARI) {a(n)=n!*polcoeff(x/(1 - x*cosh(x+x*O(x^n)))^n/n, n)}
%Y Cf. A215364, A213643.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Aug 08 2012
| 