OFFSET
1,2
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.
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.
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.
EXAMPLE
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)
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 *)
PROG
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Peter Bala, Jul 13 2007
EXTENSIONS
More terms from Petros Hadjicostas, Jul 25 2020
STATUS
approved