%I A009006
%S A009006 1,1,0,2,0,16,0,272,0,7936,0,353792,0,22368256,0,1903757312,0,
%T A009006 209865342976,0,29088885112832,0,4951498053124096,0,
%U A009006 1015423886506852352,0,246921480190207983616,0
%N A009006 Expansion of 1+tan(x).
%C A009006 If b(0)=1 and b(n+1) = -sum(u(k)*binomial(n,k)*2^(n-k-1),k=0..n-1) then 
               a(n) = abs(b(n)) (in fact b(n) = 1,1,0,-2,0,16,0,-272,...). - Robert 
               FERREOL (ferreol(AT)mathcurve.com), Dec 30 2006
%C A009006 Sum_{k, 0<=k<=n}A075263(n,k)*2^k = 1,-1,0,2,0,-16,0,272,0,-7936,0,...for 
               n=0, 1, 2, 3, 4, ...respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Aug 20 2007
%D A009006 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1997; See 
               Exercise 1.41(d).
%H A009006 Kwang-Wu Chen, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               An Interesting Lemma for Regular C-fractions</a>, J. Integer Seqs., 
               Vol. 6, 2003.
%F A009006 Let b(n) be a(n) shifted one place to the left with b(2+4k)=-a(3+4k), 
               k=0, 1, .. Then b(n) is the expansion of sech(x)^2. - Mario Catalani 
               (mario.catalani(AT)unito.it), Feb 08 2003
%F A009006 g(x)=x+x^2-2*x^4+16*x^6-272*x^8+... satisfies g(x/(1+2x))=-g(-x).
%F A009006 E.g.f.: 1+tan(x).
%F A009006 E.g.f. exp(x)sech(x) is 1,1,0,-2,0,16,0,-272,... - Paul Barry (pbarry(AT)wit.ie), 
               Mar 15 2006
%F A009006 a(n)= 2^n*abs(Euler(n,0)) where Euler(n,x) is the n-th Eulerian polynomial. 
               - Robert FERREOL (ferreol(AT)mathcurve.com), Dec 30 2006
%p A009006 u:=proc(n) if n=0 then 1 else -add(u(k)*binomial(n,k)/2*2^(n-k),k=0..n-1) 
               fi end;seq(u(n),n=0..15); - Robert FERREOL (ferreol(AT)mathcurve.com), 
               Dec 30 2006
%t A009006 1+Tan[ x ]
%t A009006 a[m_] := Abs[Sum[(-2)^(m-k) k! StirlingS2[m,k], {k,0,m}]]; Table[a[i], 
               {i,0,20}] [From Peter Luschny (peter(AT)luschny.de), Apr 29 2009]
%o A009006 (PARI) a(n)=if(n<1,n==0,n!*polcoeff(tan(x+x*O(x^n)),n))
%Y A009006 A000182(n)=a(2n-1).
%Y A009006 Sequence in context: A111978 A146558 A025600 this_sequence A155585 A057375 
               A009045
%Y A009006 Adjacent sequences: A009003 A009004 A009005 this_sequence A009007 A009008 
               A009009
%K A009006 nonn
%O A009006 0,4
%A A009006 R. H. Hardin (rhhardin(AT)att.net)
%E A009006 Reformatted Mar 15 1997