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A000828
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E.g.f. cos(x)*(cos(x)+sin(x)) /cos(2*x).
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9
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1, 1, 2, 8, 40, 256, 1952, 17408, 177280, 2031616, 25866752, 362283008, 5535262720, 91620376576, 1633165156352, 31191159799808, 635421069967360, 13753735117275136, 315212388819402752, 7625476699018231808
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Expansion of 1/(1- tan x) - Emeric Deutsch, Sep 10 2001
a(n) = A000831(n)/2 for n>0.[Peter Luschny, Nov 25 2010]
For a refinement of these numbers see A185896.
A signed permutation is a sequence (x_1,x_2,...,x_n) of integers such that {|x_1|,|x_2|,...|x_n|} = {1,2...,n}. Let x_1,...,x_n be a signed permutation. Then we say 0,x_1,...,x_n,0 is a snake of type S(n;0,0) when 0 < x_1 > x_2 < ... 0. For example, 0 4 -3 -1 -2 0 is a snake of type S(4;0,0). Then a(n) equals the cardinality of S(n;0,0) [Verges]. An example is given below. - Peter Bala, Sept 02 2011
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REFERENCES
| D. Dumont, Further triangles of Seidel-Arnold type and continued fractions related to Euler and Springer numbers, Adv. Appl. Math., 16 (1995), 275-296.
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LINKS
| R. J. Mathar, Table of n, a(n) for n = 1..200
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565
J-Verges, Enumeration of snakes and cycle-alternating permutations, arXiv:1011.0929v1 [math.CO]
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FORMULA
| a(n)=sum(evenp(n+k), k=1..n, (-1)^((n+k)/2)*sum(j=k..n, j!/n!*stirling2(n,j)*2^(n-j)*(-1)^(n+j-k)*binomial(j-1,k-1)), n>0 [From Kruchinin Vladimir, Aug 18 2010]
a(n) = (-1)^((n^2-n)/2)*4^n*(E_{n}(1/2)+E_{n}(1))/2 for n >= 0, where E_{n}(x) is an Euler polynomial. [Peter Luschny, Nov 25 2010]
a(n) = (2*I)^(n-1)*sum {k = 1..n} (-1)^(n-k)*k!* Stirling2(n,k) *
((1-I)/2)^(k-1), where I = sqrt(-1).
a(n) = 2^(n-1)*A000111(n) for n >= 1.
Let f(x) = 1+x^2 and define the effect of the operator D on a function g(x) by D(g(x)) = d/dx(f(x)*g(x)). Then for n >= 0, a(n+1) = D^n(1) evaluated at x = 1. - Peter Bala, Sept 02 2011
G.f.: 1/(1 - tan(x)) = 1 + x/(G(0) - x) ; G(k) = 2*k + 1 - (x^2)/G(k+1) ; (continued fraction ). - Sergei N. Gladkovskii, Dec 09 2011
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EXAMPLE
| a(2) = 8: The eight snakes of type S(3;0,0) are
0 1 -2 3 0, 0 1 -3 2 0, 0 2 1 3 0, 0 2 -1 3 0, 0 2 -3 1 0,
0 3 1 2 0, 0 3 -1 2 0, 0 3 -2 1 0.
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MAPLE
| A000828 := n -> (-1)^((n-1)*n/2)*4^n*(Euler(n, 1/2)+Euler(n, 1))/2: [Peter Luschny, Nov 25 2010]
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PROG
| (Maxima) a(n):=sum(if evenp(n+k) then (-1)^((n+k)/2)*sum(j!/n!*stirling2(n, j)*2^(n-j)*(-1)^(n+j-k)*binomial(j-1, k-1), j, k, n), k, 1, n); [From Kruchinin Vladimir , Aug 18 2010]
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CROSSREFS
| Cf. A000825. A000111, A185896
Sequence in context: A116456 A055882 A002301 * A180736 A111394 A140363
Adjacent sequences: A000825 A000826 A000827 * A000829 A000830 A000831
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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