A131455 - OEIS

%I #35 Mar 07 2022 07:55:12

%S 1,2,18,284,7280,273246,14144592,965491288,84027112704,9081387766810,

%T 1193283000239616,187340544144604212,34633340434838499328,

%U 7446726867419368499894,1842612127654047957411840,519870106084045866346942256,165896395346243470375430193152,59450668490817059243377908811698,23773400714993519201980928470155264

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

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

%C 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.

%H Alois P. Heinz, <a href="/A131455/b131455.txt">Table of n, a(n) for n = 1..250</a>

%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]

%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.

%H B. Shapiro and A. Vainshtein, <a href="http://arXiv.org/abs/math.AG/0209062">Counting real rational functions with all real critical values</a>, arXiv:math/0209062 [math.AG], 2002.

%H B. Shapiro and A. Vainshtein, <a href="http://www.mathnet.ru/php/archive.phtml?wshow=paper&amp;jrnid=mmj&amp;paperid=103&amp;option_lang=eng">Counting real rational functions with all real critical values</a>, Moscow Math. J., 3 (2003), 647-659.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GeneralizedHyperbolicFunctions.html">Generalized Hyperbolic Functions</a>.

%F 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.

%e From _Petros Hadjicostas_, Jul 25 2020: (Start)

%e For n = 2, the a(2)/2 = 1 "2 up, 2 down" permutation of length 2*2 - 1 = 3 is the following:

%e          3

%e         /

%e        2

%e       /

%e      1

%e For n = 3, the a(3)/3 = 6 "2 up, 2 down" permutations of length 2*3 - 1 = 5 are the following:

%e         5          5          5          5          5          5

%e        / \        / \        / \        / \        / \        / \

%e       3   4      4   3      2   4      3   4      4   3      4   2

%e      /     \    /     \    /     \    /     \    /     \    /     \

%e     1       2  1       2  1       3  2       1  2       1  3       1

%e (End)

%p b:= proc(u, o, t) option remember; `if`(u+o=0, 1, add(

%p      `if`(t=2, b(o-j, u+j-1, 1), b(u+j-1, o-j, t+1)), j=1..o))

%p     end:

%p a:= n-> n*b(0, 2*n-1, 0):

%p seq(a(n), n=1..19);  # _Alois P. Heinz_, Nov 23 2021

%t b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, Sum[If[t == 2,

%t      b[o - j, u + j - 1, 1], b[u + j - 1, o - j, t + 1]], {j, 1, o}]];

%t a[n_] := n*b[0, 2*n - 1, 0];

%t Table[a[n], {n, 1, 19}] (* _Jean-François Alcover_, Mar 07 2022, after _Alois P. Heinz_ *)

%o (PARI) f(j,x,nn) = sum(k=0, 2*nn, (x^(4*k + j))/(4*k + j)!);

%o 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));

%o 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

%Y Cf. A005981, A131453, A131454.

%K easy,nonn

%O 1,2

%A _Peter Bala_, Jul 13 2007

%E More terms from _Petros Hadjicostas_, Jul 25 2020