A131453 -id:A131453 - cp's OEIS frontend

A005981 Number of 2 up, 2 down, 2 up, ... permutations of length 2n + 1.

1, 1, 6, 71, 1456, 45541, 2020656, 120686411, 9336345856, 908138776681, 108480272749056, 15611712012050351, 2664103110372192256, 531909061958526321421, 122840808510269863827456, 32491881630252866646683891, 9758611490955498257378246656

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
P. R. Stein, personal communication.

Alois P. Heinz, Table of n, a(n) for n = 0..100
Nicolas Basset, Counting and generating permutations using timed languages, 2013.
Nicolas Basset, Counting and generating permutations in regular classes of permutations, HAL Id: hal-01093994, 2014.
B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, arXiv:math/0209062 [math.AG], 2002; Moscow Math. J., 3 (2003), 647-659.
P. R. Stein & N. J. A. Sloane, Correspondence, 1975
Eric Weisstein's World of Mathematics, Generalized Hyperbolic Functions

Bisection of A058258.

Cf. A131453, A131454, A131455, A076417.

g:=((cosh(x)-1)*sin(x)+(cos(x)+1)*sinh(x))/(cos(x)*cosh(x)+1): series(%,x,35):
seq(n!*coeff(%,x,n),n=1..34,2); # Peter Luschny, Feb 07 2017

egf = ((Cosh[x]-1)*Sin[x]+(Cos[x]+1)*Sinh[x])/(Cos[x]*Cosh[x]+1); a[n_] := SeriesCoefficient[egf, {x, 0, 2*n+1}]*(2*n+1)!; Array[a, 17, 0] (* Jean-François Alcover, Mar 13 2014 *)

E.g.f.: x + Sum_{n>=1} a(n)*(x^(2n+1))/(2n+1)! = (f(0,x)*f(1,x) -f(2,x)*f(3,x)+ f(3,x))/(f(0,x)^2 - f(1,x)*f(3,x)), where f(j,x) = Sum_{k>=0} (x^(4k+j))/(4k+j)!, j = 0,1,2,3, is the j-th generalized hyperbolic function. - Peter Bala, Jul 13 2007
Basset (2013) gives an e.g.f. involving trigonometric and hyperbolic functions. - N. J. A. Sloane, Dec 24 2013
a(n) ~ 4 * (2*n+1)! / (tan(r/2)^2 * r^(2*n+2)), where r = A076417 = 1.8751040687119611664453082410782141625701117335310699882454137131... is the root of the equation cos(r)*cosh(r) = -1. - Vaclav Kotesovec, Feb 01 2015

A131454 2 up, 2 down, ..., 2 up, 2 down, 2 up permutations of length 4n+3.

Original entry on oeis.org

1, 71, 45541, 120686411, 908138776681, 15611712012050351, 531909061958526321421, 32491881630252866646683891, 3302814916156503291298772711761, 527393971346575736206847604137659031, 126355819963625435928020023737689391659701

Offset: 0

Peter Bala, Jul 13 2007

Bisection of A005981. The entries listed above suggest various congruences for a(n): a(n) = 1 (mod 10), a(n) = 1 + 70*n (mod 100), a(n) = 1 + 70*n + 200*n(n-1) (mod 1000). Are these congruences true for all n? For an arbitrary integer m, the sequence a(n) taken modulo m may eventually become periodic. Compare with A081727.

			(1 4 5 3 2 6 7) is a 2 up, 2 down, 2 up permutation - one of the seventy-one permutations of this type in the symmetric group on 7 letters.

Alois P. Heinz, Table of n, a(n) for n = 0..50
Christopher R. H. Hanusa, Alejandro H. Morales, and Martha Yip, Column convex matrices, G-cyclic orders, and flow polytopes, arXiv:2107.07326 [math.CO], 2021.
L. Olivier, Bemerkungen über eine Art von Functionen, welche ähnliche Eigenschaften haben, wie der Cosinus und Sinus, J. Reine Angew. Math. 2 (1827), 243-251.
H. Prodinger and T. A. Tshifhumulo, On q-Olivier Functions, Annals of Combinatorics 6 (2002), 181-194.
B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, arXiv:math/0209062 [math.AG], 2002.
B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, Moscow Math. J. 3 (2003), 647-659.

Cf. A000111, A005981, A058257, A131453, A131455.

g:=(sinh(x)-sin(x))/(cos(x)*cosh(x)+1): series(%,x,44):
seq(n!*coeff(%,x,n),n=3..45,4); # Peter Luschny, Feb 07 2017

Table[(CoefficientList[Series[(-Sin[x] + Sinh[x]) / (1 + Cos[x]*Cosh[x]), {x, 0, 60}], x] * Range[0, 59]!)[[n]], {n, 4, 60, 4}] (* Vaclav Kotesovec, Sep 09 2014 *)

E.g.f.: Sum_{n>=0} a(n)*(x^(4n+3))/(4n+3)! = (exp(2x)-2*sin(x)*exp(x)-1)/(2*exp(x)+cos(x)*(exp(2x)+1)). It appears that a(n) = (4n+3)!*coefficient of x^(4n+3) in the Taylor expansion of -4/(2*exp(x)+cos(x)*(exp(2x)+1)).

A131455 Number of inequivalent properly oriented and labeled planar chord diagrams whose associated planar tree is a path on n + 1 vertices.

Original entry on oeis.org

1, 2, 18, 284, 7280, 273246, 14144592, 965491288, 84027112704, 9081387766810, 1193283000239616, 187340544144604212, 34633340434838499328, 7446726867419368499894, 1842612127654047957411840, 519870106084045866346942256, 165896395346243470375430193152, 59450668490817059243377908811698, 23773400714993519201980928470155264

Offset: 1

Peter Bala, Jul 13 2007

a(n) = n times the number of "2 up, 2 down" permutations of length 2*n-1 = n*A005981(n-1) for n >= 2.

a(n) ~ (c_1)*n*(2*n - 1)!/(c_2)^(2n), where c_1 is a constant and c_2 = 1.87510... is the smallest positive solution of the equation cos(z)* cosh(z) + 1 = 0.

			From _Petros Hadjicostas_, Jul 25 2020: (Start)
For n = 2, the a(2)/2 = 1 "2 up, 2 down" permutation of length 2*2 - 1 = 3 is the following:
         3
        /
       2
      /
     1
For n = 3, the a(3)/3 = 6 "2 up, 2 down" permutations of length 2*3 - 1 = 5 are the following:
        5          5          5          5          5          5
       / \        / \        / \        / \        / \        / \
      3   4      4   3      2   4      3   4      4   3      4   2
     /     \    /     \    /     \    /     \    /     \    /     \
    1       2  1       2  1       3  2       1  2       1  3       1
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..250
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, arXiv:math/0209062 [math.AG], 2002.
B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, Moscow Math. J., 3 (2003), 647-659.
Eric Weisstein's World of Mathematics, Generalized Hyperbolic Functions.

Cf. A005981, A131453, A131454.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1, add(
     `if`(t=2, b(o-j, u+j-1, 1), b(u+j-1, o-j, t+1)), j=1..o))
    end:
a:= n-> n*b(0, 2*n-1, 0):
seq(a(n), n=1..19);  # Alois P. Heinz, Nov 23 2021

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, Sum[If[t == 2,
     b[o - j, u + j - 1, 1], b[u + j - 1, o - j, t + 1]], {j, 1, o}]];
a[n_] := n*b[0, 2*n - 1, 0];
Table[a[n], {n, 1, 19}] (* Jean-François Alcover, Mar 07 2022, after Alois P. Heinz *)

f(j,x,nn) = sum(k=0, 2*nn, (x^(4*k + j))/(4*k + j)!);
g(x,nn) = (x/2)*(f(0,x,nn)*f(1,x,nn) - f(2,x,nn)*f(3,x,nn) + f(3,x,nn))/(f(0,x,nn)^2 - f(1,x,nn)*f(3,x,nn));
lista(nn) = {default(seriesprecision, 2*nn); my(a=vector(nn)); for(n=1, nn, a[n] = (2*n)!*polcoef(Ser(g(x,nn)), 2*n)); a;} \\ Petros Hadjicostas, Jul 25 2020

E.g.f.: Sum_{n >= 1} a(n)*(x^(2*n))/(2*n)! = (x/2)*(f(0,x)*f(1,x) - f(2,x)*f(3,x) + f(3,x))/(f(0,x)^2 - f(1,x)*f(3,x)), where f(j,x) = Sum_{k >= 0} (x^(4*k + j))/(4*k + j)!, j = 0, 1, 2, 3, is the j-th generalized hyperbolic function.

More terms from Petros Hadjicostas, Jul 25 2020

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A005981 Number of 2 up, 2 down, 2 up, ... permutations of length 2n + 1.

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A131454 2 up, 2 down, ..., 2 up, 2 down, 2 up permutations of length 4n+3.

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A131455 Number of inequivalent properly oriented and labeled planar chord diagrams whose associated planar tree is a path on n + 1 vertices.

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