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A009006 Expansion of e.g.f.: 1 + tan(x). 19

%I

%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).

%C If b(0)=1 and b(n+1) = -Sum_{k=0..n-1} u(k)*binomial(n,k)*2^(n-k-1) then a(n) = abs(b(n)) (in fact, b(n) = 1,1,0,-2,0,16,0,-272,...). - _Robert FERREOL_, Dec 30 2006

%C 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

%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 a(n) = 2^n*abs(Euler(n,0)) where Euler(n,x) is the n-th Eulerian polynomial. - _Robert FERREOL_, Dec 30 2006

%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

%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 *)

%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

%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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Last modified October 24 01:20 EDT 2018. Contains 316541 sequences. (Running on oeis4.)