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A000828
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E.g.f. cos(x)/(cos(x) - sin(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
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OFFSET
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0,3
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COMMENTS
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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
Original name was: E.g.f. cos(x)*(cos(x)+sin(x)) /cos(2*x). [Arkadiusz Wesolowski, Jul 25 2012]
Number of plane (that is, ordered) increasing 0-1-2 trees on n vertices where the vertices of outdegree 1 or 2 come in two colors. An example is given below. - Peter Bala, Oct 10 2012
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REFERENCES
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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
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R. J. Mathar, Table of n, a(n) for n = 1..200
F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1922, pp. 24-48.
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565
M. Josuat-Verges, Enumeration of snakes and cycle-alternating permutations, arXiv:1011.0929v1 [math.CO]
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FORMULA
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E.g.f.: 1/(1- tan(x)) - Emeric Deutsch, Sep 10 2001
a(n) = A000831(n)/2 for n>0. [Peter Luschny, Nov 25 2010]
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 Vladimir Kruchinin, 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 + x/(G(0) - x) ; G(k) = 2*k + 1 - (x^2)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 09 2011
G.f.: 1 / (1 - 1*x / (1 - 1*x / (1 - 4*x / (1 - 2*6*x^2 / (1 - 6*x / (1 - 4*x / (1 - 4*x / (1 - 10*x / (1 - 5*12*x^2 / (1 - 12*x / ...)))))))))). - Michael Somos, May 12 2012
E.g.f.: 1+x/(U(0)-2*x) where U(k)= 4*k+1 + x/(1+x/(4*k+3 - x/(1- x/U(k+1)))); (continued fraction, 4-step). - Sergei N. Gladkovskii, Nov 08 2012
E.g.f.: 1+x/(U(0)-x) where U(k)= 2*k+1 - x^2/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Dec 04 2012
G.f.: 1 + x/G(0) where G(k) = 1 - x*(2*k+2) - 2*x^2*(k+1)*(k+2)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 11 2013.
E.g.f.: 1 + x/T(0) where T(k) = 4*k+1 - x/(1 - x/(4*k+3 + x/(1 + x/T(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Mar 17 2013
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EXAMPLE
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a(3) = 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.
1 + x + 2*x^2 + 8*x^3 + 40*x^4 + 256*x^5 + 1952*x^6 + 17408*x^7 + ...
a(3) = 8: The eight increasing 0-1-2 trees on 3 vertices are
..1o (x2 colors)......1o (x2 colors)......1o (x2 colors).....
...|................./.\................./.\.................
..2o (x2 colors)...2o...o3.............3o...o2...............
...|
..3o
Totals.......................................................
...4......+...........2.........+.........2....=...8.........
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MAPLE
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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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MATHEMATICA
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a[n_] := (-1)^((n-1)*n/2)*4^n*(EulerE[n, 1/2] + EulerE[n, 1])/2; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Nov 22 2012, after Peter Luschny *)
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PROG
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(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 Vladimir Kruchinin, Aug 18 2010]
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CROSSREFS
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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
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nonn
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Name changed by Arkadiusz Wesolowski, Jul 25 2012
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STATUS
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approved
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