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%I #30 Sep 12 2015 11:00:27
%S 1,1,3,14,90,736,7308,85364,1146660,17411296,294880608,5510730224,
%T 112638576960,2499645858256,59850581734128,1537891759461344,
%U 42211161452391840,1232537502720495616,38148561442665067968,1247578217573259535424,42985394533206479112000
%N E.g.f. satisfies: A(x) = 1+x - cos(A(x)).
%C Radius of convergence of e.g.f. A(x) is r = Pi/2-1 = 0.570796... where A(r) = Pi/2.
%H Vaclav Kotesovec, <a href="/A200317/b200317.txt">Table of n, a(n) for n = 1..410</a>
%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Asymptotic of implicit functions if Fww = 0</a>, Jan 19 2014
%F E.g.f. satisfies:
%F (1) A(x) = Series_Reversion(x-1 + cos(x)).
%F (2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (1-cos(x))^(2*n) / n!.
%F (3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (1-cos(x))^(2*n)/x / n! ).
%F a(n) = ((n-1)!*sum(k=1..n-1, C(n+k-1,n-1)* sum(j=1..k, C(k,j)* ((sum(l=0..j-1, (C(j,l)* ((-1)^(n-l+j-1)+1) *sum(r=1..j-l, (C(j-l,r)*(-1)^(-r+(n-l+j-1)/2-l)* sum(i=0..(r-1)/2, (r-2*i)^(n-l+j-1)*C(r,i)))/2^r))/(n-l+j-1)!)))))), n>1, a(1)=1. [From Vladimir Kruchinin, Feb 20 2012]
%F a(n) ~ GAMMA(1/3) * n^(n-5/6) / (6^(1/6) * sqrt(Pi) * exp(n) * (Pi/2-1)^(n-1/3)). - _Vaclav Kotesovec_, Jan 18 2014
%e E.g.f.: A(x) = x + x^2/2! + 3*x^3/3! + 14*x^4/4! + 90*x^5/5! +...
%e where A(x-1 + cos(x)) = x and A(x) = 1+x - cos(A(x)).
%e The e.g.f. satisfies:
%e A(x) = x + (1-cos(x)) + d/dx (1-cos(x))^2/2! + d^2/dx^2 (1-cos(x))^3/3! + d^3/dx^3 (1-cos(x))^4/4! +...
%e as well as the logarithmic series:
%e log(A(x)/x) = (1-cos(x))/x + d/dx (1-cos(x))^2/x/2! - d^2/dx^2 (1-cos(x))^3/x/3! + d^3/dx^3 (1-cos(x))^4/x/4! +...
%t Rest[CoefficientList[InverseSeries[Series[x-1 + Cos[x], {x, 0, 20}], x],x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Jan 18 2014 *)
%o (PARI) {a(n)=n!*polcoeff(serreverse(x-1+cos(x+x^2*O(x^n))),n)}
%o for(n=1, 21, 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+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, (1-cos(x+x*O(x^n)))^m)/m!)+x*O(x^n)); 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)); for(i=1, n, A=x*exp(sum(m=1, n, Dx(m-1, (1-cos(x+x*O(x^n)))^m/x)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
%o (Maxima) a(n):=if n=1 then 1 else ((n-1)!*sum(binomial(n+k-1,n-1)* sum(binomial(k,j)*((sum((binomial(j,l)*((-1)^(n-l+j-1)+1)*sum((binomial(j-l,r) *(-1)^(-r+(n-l+j-1)/2-l)*sum((r-2*i)^(n-l+j-1)*binomial(r,i),i,0,(r-1)/2))/2^r,r,1,j-l))/(n-l+j-1)!,l,0,j-1))),j,1,k),k,1,n-1)); [From Vladimir Kruchinin, Feb 20 2012]
%Y Cf. A200318.
%K nonn,nice
%O 1,3
%A _Paul D. Hanna_, Nov 15 2011
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