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%I #12 Jul 16 2014 16:10:27
%S 0,1,2,9,72,825,12192,220353,4708480,116116497,3245839360,
%T 101415497689,3502465714176,132486192976137,5447446920323072,
%U 241907419042038225,11538444129924055040,588321821566662253729,31932991994214557417472
%N E.g.f. satisfies: A(x) = x*(exp(tan(A(x)))).
%H Alois P. Heinz, <a href="/A133941/b133941.txt">Table of n, a(n) for n = 0..100</a>
%F a(n) = sum(k=1..n-1, (n^k*((-1)^(n-k-1)+1)*sum(j=k..n-1, C(j-1,k-1)*j!*2^(n-j-2) * (-1)^((n+k-1)/2+j)*stirling2(n-1,j),j,k,n-1))/k!) n>1. a(1)=1. - Vladimir Kruchinin, May 10 2011
%F a(n) ~ n^(n-1) * s / (exp(n) * r^n * sqrt(1+sin(2*s))), where r = 0.3039707088650079908... and s = 0.6417143708728826584... are roots of the system of equations exp(tan(s))*r = s, s = (cos(s))^2. - _Vaclav Kotesovec_, Jul 16 2014
%p A:= proc(n) option remember; if n=0 then 0 else convert (series (x* (exp (tan(A(n-1)))), x=0, n+1), polynom) fi end: a:= n-> coeff (A(n), x, n)*n!: seq (a(n), n=0..22);
%t CoefficientList[InverseSeries[Series[x/E^Tan[x],{x,0,20}],x],x] * Range[0,20]! (* _Vaclav Kotesovec_, Jul 16 2014 *)
%o (Maxima) a(n):=if n<2 then n else sum((n^k*((-1)^(n-k-1)+1)*sum(binomial(j-1,k-1)*j!*2^(n-j-2)*(-1)^((n+k-1)/2+j)*stirling2(n-1,j),j,k,n-1))/k!,k,1,n-1) [Vladimir Kruchinin, May 10 2011]
%K nonn
%O 0,3
%A _Alois P. Heinz_, Aug 27 2008
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