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%I M4301 #39 Oct 29 2022 16:07:10
%S 1,6,168,10672,1198080,208521728,51874413568,17449541107712,
%T 7622674735988736,4193561606973095936,2836052065377836597248,
%U 2312174256451088534208512,2236165580390456719589769216,2530976708469616321520834969600,3314110602212685014002135203840000
%N Expansion of e.g.f. tan(tan(tan(x))).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vladimir Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%F a(n) = b(2*n+1) where b(n) = sum(m=1..n, (((-1)^(m-1)+1)*(sum(j=1..m, j! *2^(m-j-1)*(-1)^((m+1)/2+j)*S2(m,j)))*sum(k=m..n,(((-1)^(k-m)+1)*(sum(j=m..k, C(j-1,m-1)*j!*2^(k-j-1)*S2(k,j)*(-1)^((m+k)/2+j)))*((-1)^(n-k)+1)* sum(j=k,n, C(j-1,k-1)*j!*2^(n-j-1)* (-1)^((n+k)/2+j)* S2(n,j)))/k!))/m!). - _Vladimir Kruchinin_, Apr 23 2011
%F a(n) ~ 8*(2*n+1)!/((4+Pi^2) * (1+arctan(Pi/2)^2) * (arctan(arctan(Pi/2)))^(2*n+2)). - _Vaclav Kotesovec_, Feb 16 2015
%t Rest@ Union[ Range[0, 25]! CoefficientList[ Series[Tan@ Tan@ Tan@ x, {x, 0, 25}], x]] (* _Robert G. Wilson v_ *)
%o (Maxima) a(n):=b(2*n+1); b(n):=sum((((-1)^(m-1)+1)*(sum(j!*2^(m-j-1)* (-1)^((m+1)/2+j) *stirling2(m,j), j,1,m))*sum((((-1)^(k-m)+1)*(sum(binomial(j-1,m-1)* j!*2^(k-j-1)*stirling2(k,j)*(-1)^((m+k)/2+j), j,m,k))*((-1)^(n-k)+1)* sum(binomial(j-1,k-1)*j!*2^(n-j-1)* (-1)^((n+k)/2+j)* stirling2(n,j) ,j,k,n))/k!,k,m,n))/m!, m,1,n); [_Vladimir Kruchinin_, Apr 23 2011]
%o (PARI) x='x+O('x^66); /* that many terms */
%o serlaplace(tan(tan(tan(x)))) /* show terms */ /* _Joerg Arndt_, Apr 26 2011 */
%K nonn
%O 0,2
%A _R. H. Hardin_
%E Extended and formatted Mar 15 1997 by _Olivier Gérard_
%E Corrected definition, _Joerg Arndt_, Apr 26 2011
%E a(13)-a(14) from _Alois P. Heinz_, May 13 2012
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