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%I #99 Mar 08 2024 16:13:59
%S 1,1,0,2,0,16,0,272,0,7936,0,353792,0,22368256,0,1903757312,0,
%T 209865342976,0,29088885112832,0,4951498053124096,0,
%U 1015423886506852352,0,246921480190207983616,0,70251601603943959887872,0,23119184187809597841473536,0
%N Expansion of e.g.f.: 1 + tan(x).
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1997; See Exercise 1.41(d).
%H G. C. Greubel, <a href="/A009006/b009006.txt">Table of n, a(n) for n = 0..451</a>
%H Kwang-Wu Chen, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Chen/chen50.html">An Interesting Lemma for Regular C-fractions</a>, J. Integer Seqs., Vol. 6, 2003.
%F 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 g(x) = x + x^2 - 2*x^4 + 16*x^6 - 272*x^8 + ... satisfies g(x/(1+2x)) = -g(-x).
%F E.g.f.: 1 + tan(x).
%F E.g.f. exp(x)*sech(x) is 1,1,0,-2,0,16,0,-272,... (A155585). - _Paul Barry_, Mar 15 2006
%F From _Robert FERREOL_, Dec 30 2006: (Start)
%F a(n) = 2^n*abs(Euler(n,0)) where Euler(n,x) is the n-th Eulerian polynomial.
%F a(n) = abs(u(n)) where u(n) = -Sum_{k=0..n-1} u(k)*binomial(n, k)*2^(n-k-1) with u(0) = 1. (End)
%F Sum_{k=0..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 Deléham_, Aug 20 2007
%F E.g.f. -log(cos(x)), for n > 0. - _Vladimir Kruchinin_, Aug 09 2010
%F a(n) = Sum_{k=1..n} Sum_{j=0..k} (-1)^(floor(n/2)+j+1)*binomial(n+1,k-j)*j^n for n > 0. - _Peter Luschny_, Jul 23 2012
%F From _Sergei N. Gladkovskii_, Oct 25 2012 - Dec 20 2013: (Start)
%F Continued fractions:
%F G.f.: 1 + x/T(0) where T(k) = 1 - (k+1)*(k+2)*x^2/T(k+1).
%F E.g.f.: 1 + tan(x) = 1+x/(U(0)-x) where U(k)= 4*k+1 + x/(1+x/(4*k+3 - x/(1- x/U(k+1)))).
%F E.g.f.: 1+tan(x) = 1 - 3*x/((U(0) + 3*x^2) where U(k) = 64*k^3 + 48*k^2 - 4*k*(2*x^2+1) - 2*x^2 - 3 - x^4*(4*k-1)*(4*k+7)/U(k+1).
%F E.g.f.: 1+x*G(0) where G(k) = 1 - x^2/(x^2 - (2*k+1)*(2*k+3)/G(k+1)).
%F G.f.: 1 + x/G(0) where G(k) = 1 - 2*x^2*(4*k^2+4*k+1)-4*x^4*(k+1)^2*(4*k^2+8*k+3) /G(k+1).
%F G.f.: 1 + x*Q(0) where Q(k) = 1 - x^2*(k+1)*(k+2)/(x^2*(k+1)*(k+2) - 1/Q(k+1)).
%F G.f.: Q(0) where Q(k) = 1 + x*(k+1)/(x*(k+1)+1/(1- x*(k+1)/(x*(k+1) - 1/Q(k+1)))).
%F E.g.f.: 2 - 1/Q(0) where Q(k) = 1 + x/(4*k+1 - x/(1 - x/(4*k+3 + x/Q(k+1)))). (End)
%F a(n) ~ 2*n!*(2/Pi)^(n+1) if n is odd. - _Vaclav Kotesovec_, Jun 01 2013
%F a(n) = i^(n+1) * 2^n * ((-1)^n-1) * (2^(n+1)-1) * Bernoulli(n+1)/(n+1), n > 0. - _Benedict W. J. Irwin_, May 27 2016
%F a(0) = a(1) = 1; a(n) = -2 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k) * a(n-k-1). - _Ilya Gutkovskiy_, Jul 05 2020
%p 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_, Dec 30 2006
%t a[m_] := Abs[Sum[(-2)^(m-k) k! StirlingS2[m,k], {k,0,m}]]; Table[a[i], {i,0,20}] (* _Peter Luschny_, Apr 29 2009 *)
%t A009006[n_] := Cos[Pi (n-1) / 2] (4^(n+1) - 2^(n+1)) * BernoulliB[n+1] / (n+1); a[0] := 1; Table[A009006[n], {n, 0, 30}] (* _Peter Luschny_, Jun 14 2021 *)
%o (PARI) a(n)=if(n<1,n==0,n!*polcoeff(tan(x+x*O(x^n)),n))
%o (Sage)
%o def A009006(n) :
%o if n == 0 : return 1
%o return add(add((-1)^(n//2+j+1)*binomial(n+1,k-j)*j^n for j in (0..k)) for k in (1..n))
%o [A009006(n) for n in (0..26)] # _Peter Luschny_, Jul 23 2012
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1 + Tan(x))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Jul 21 2018
%Y A000182(n) = a(2n-1).
%K nonn,easy
%O 0,4
%A _R. H. Hardin_
%E Reformatted Mar 15 1997
%E Definition corrected by _Joerg Arndt_, Apr 29 2011
%E Terms a(26) onward added by _G. C. Greubel_, Jul 21 2018
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