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A000828 E.g.f. cos(x)/(cos(x) - sin(x)). 19
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; text; internal format)
OFFSET

0,3

COMMENTS

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, Sep 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

LINKS

R. J. Mathar, Table of n, a(n) for n = 0..200

F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.

D. Dumont, Further triangles of Seidel-Arnold type and continued fractions related to Euler and Springer numbers, Adv. Appl. Math., 16 (1995), 275-296.

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]

FORMULA

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. - 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, Sep 02 2011

From Sergei N. Gladkovskii, Dec 09 2011 - Dec 23 2013: (Start) Continued fractions:

E.g.f.: 1 + x/(G(0)-x); G(k) = 2*k + 1 - (x^2)/G(k+1).

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)))).

E.g.f.: 1 + x/(U(0)-x) where U(k) = 2*k+1 - x^2/U(k+1).

G.f.: 1 + x/G(0) where G(k) = 1 - x*(2*k+2) - 2*x^2*(k+1)*(k+2)/G(k+1).

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)))).

G.f.: 1 + x/Q(0) where Q(k) = 1 - 2*x*(2*k+1) - 2*x^2*(2*k+1)*(2*k+2)/(1 - 2*x*(2*k+2) - 2*x^2*(2*k+2)*(2*k+3)/Q(k+1)).

G.f.: 1 + x/(1-2*x)*T(0) where T(k) = 1 - 2*x^2*(k+1)*(k+2)/( 2*x^2*(k+1)*(k+2) - (1 - 2*x*(k+1))*(1 - 2*x*(k+2))/T(k+1)).

E.g.f.: T(0) where T(k) = 1 + x/(4*k+1 - x/(1 - x/( 4*k+3 + x/T(k+1)))). (End)

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

a(n) ~ n! * 2^(2*n+1)/Pi^(n+1). - Vaclav Kotesovec, Jun 21 2013

EXAMPLE

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.........

MAPLE

A000828 := n -> (-1)^((n-1)*n/2)*4^n*(Euler(n, 1/2)+Euler(n, 1))/2: # Peter Luschny, Nov 25 2010

MATHEMATICA

a[n_] := (-1)^((n-1)*n/2)*4^n*(EulerE[n, 1/2] + EulerE[n, 1])/2; Table[a[n], {n, 0, 19}] (* Jean-Fran├žois Alcover, Nov 22 2012, after Peter Luschny *)

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); /* Vladimir Kruchinin, Aug 18 2010 */

CROSSREFS

Cf. A000825. A000111, A185896, A235131, A235132.

Sequence in context: A304070 A259869 A321733 * A296676 A281910 A180736

Adjacent sequences:  A000825 A000826 A000827 * A000829 A000830 A000831

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

Name changed by Arkadiusz Wesolowski, Jul 25 2012

STATUS

approved

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Last modified August 18 13:25 EDT 2019. Contains 326100 sequences. (Running on oeis4.)