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A009006 Expansion of e.g.f.: 1 + tan(x). 26
1, 1, 0, 2, 0, 16, 0, 272, 0, 7936, 0, 353792, 0, 22368256, 0, 1903757312, 0, 209865342976, 0, 29088885112832, 0, 4951498053124096, 0, 1015423886506852352, 0, 246921480190207983616, 0, 70251601603943959887872, 0, 23119184187809597841473536, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
REFERENCES
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1997; See Exercise 1.41(d).
LINKS
Kwang-Wu Chen, An Interesting Lemma for Regular C-fractions, J. Integer Seqs., Vol. 6, 2003.
FORMULA
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
g(x) = x + x^2 - 2*x^4 + 16*x^6 - 272*x^8 + ... satisfies g(x/(1+2x)) = -g(-x).
E.g.f.: 1 + tan(x).
E.g.f. exp(x)*sech(x) is 1,1,0,-2,0,16,0,-272,... (A155585). - Paul Barry, Mar 15 2006
From Robert FERREOL, Dec 30 2006: (Start)
a(n) = 2^n*abs(Euler(n,0)) where Euler(n,x) is the n-th Eulerian polynomial.
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)
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
E.g.f. -log(cos(x)), for n > 0. - Vladimir Kruchinin, Aug 09 2010
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
From Sergei N. Gladkovskii, Oct 25 2012 - Dec 20 2013: (Start)
Continued fractions:
G.f.: 1 + x/T(0) where T(k) = 1 - (k+1)*(k+2)*x^2/T(k+1).
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)))).
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).
E.g.f.: 1+x*G(0) where G(k) = 1 - x^2/(x^2 - (2*k+1)*(2*k+3)/G(k+1)).
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).
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)).
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)))).
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)
a(n) ~ 2*n!*(2/Pi)^(n+1) if n is odd. - Vaclav Kotesovec, Jun 01 2013
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
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
MAPLE
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
MATHEMATICA
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 *)
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 *)
PROG
(PARI) a(n)=if(n<1, n==0, n!*polcoeff(tan(x+x*O(x^n)), n))
(Sage)
def A009006(n) :
if n == 0 : return 1
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))
[A009006(n) for n in (0..26)] # Peter Luschny, Jul 23 2012
(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
CROSSREFS
A000182(n) = a(2n-1).
Sequence in context: A146558 A364514 A025600 * A155585 A350972 A236755
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Reformatted Mar 15 1997
Definition corrected by Joerg Arndt, Apr 29 2011
Terms a(26) onward added by G. C. Greubel, Jul 21 2018
STATUS
approved

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Last modified April 24 18:17 EDT 2024. Contains 371962 sequences. (Running on oeis4.)