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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.

%I A131455 #36 Feb 16 2025 08:33:06
%S A131455 1,2,18,284,7280,273246,14144592,965491288,84027112704,9081387766810,
%T A131455 1193283000239616,187340544144604212,34633340434838499328,
%U A131455 7446726867419368499894,1842612127654047957411840,519870106084045866346942256,165896395346243470375430193152,59450668490817059243377908811698,23773400714993519201980928470155264
%N A131455 Number of inequivalent properly oriented and labeled planar chord diagrams whose associated planar tree is a path on n + 1 vertices.
%C A131455 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 A131455 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 A131455 Alois P. Heinz, <a href="/A131455/b131455.txt">Table of n, a(n) for n = 1..250</a>
%H A131455 Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H A131455 Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H A131455 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 A131455 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 A131455 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GeneralizedHyperbolicFunctions.html">Generalized Hyperbolic Functions</a>.
%F A131455 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 A131455 From _Petros Hadjicostas_, Jul 25 2020: (Start)
%e A131455 For n = 2, the a(2)/2 = 1 "2 up, 2 down" permutation of length 2*2 - 1 = 3 is the following:
%e A131455          3
%e A131455         /
%e A131455        2
%e A131455       /
%e A131455      1
%e A131455 For n = 3, the a(3)/3 = 6 "2 up, 2 down" permutations of length 2*3 - 1 = 5 are the following:
%e A131455         5          5          5          5          5          5
%e A131455        / \        / \        / \        / \        / \        / \
%e A131455       3   4      4   3      2   4      3   4      4   3      4   2
%e A131455      /     \    /     \    /     \    /     \    /     \    /     \
%e A131455     1       2  1       2  1       3  2       1  2       1  3       1
%e A131455 (End)
%p A131455 b:= proc(u, o, t) option remember; `if`(u+o=0, 1, add(
%p A131455      `if`(t=2, b(o-j, u+j-1, 1), b(u+j-1, o-j, t+1)), j=1..o))
%p A131455     end:
%p A131455 a:= n-> n*b(0, 2*n-1, 0):
%p A131455 seq(a(n), n=1..19);  # _Alois P. Heinz_, Nov 23 2021
%t A131455 b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, Sum[If[t == 2,
%t A131455      b[o - j, u + j - 1, 1], b[u + j - 1, o - j, t + 1]], {j, 1, o}]];
%t A131455 a[n_] := n*b[0, 2*n - 1, 0];
%t A131455 Table[a[n], {n, 1, 19}] (* _Jean-François Alcover_, Mar 07 2022, after _Alois P. Heinz_ *)
%o A131455 (PARI) f(j,x,nn) = sum(k=0, 2*nn, (x^(4*k + j))/(4*k + j)!);
%o A131455 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 A131455 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 A131455 Cf. A005981, A131453, A131454.
%K A131455 easy,nonn
%O A131455 1,2
%A A131455 _Peter Bala_, Jul 13 2007
%E A131455 More terms from _Petros Hadjicostas_, Jul 25 2020