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

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

Offset: 1

Author

Peter Bala, Jul 13 2007

Keywords

• easy
• nonn

Comments

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.

Examples

			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)
		

Links

• 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.

Crossrefs

Cf. A005981, A131453, A131454.

Programs

Maple

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

Mathematica

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 *)

PARI

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

Formula

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.

Extensions

More terms from Petros Hadjicostas, Jul 25 2020