OFFSET
0,4
COMMENTS
A permutation of {1,2,...,n} is said to be a "broken alternating permutation" if it is an alternating permutation (cf. A000111) except at one point. See El Hilany and Rau for precise definition and an explicit formula.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..484
D. Chebikin, Variations on descents and inversions in permutations, The Electronic J. of Combinatorics, 15 (2008), #R132.
Boulos El Hilany, Johannes Rau, Signed counts of real simple rational functions, arXiv:1712.05639 [math.AG], 2017, Proposition 6.4, p. 19.
FORMULA
a(n) ~ (4 - Pi) * 2^(n + 5/2) * n^(n + 3/2) / (exp(n) * Pi^(n + 3/2)). - Vaclav Kotesovec, Apr 14 2018
E.g.f.: (cos(x)-sin(x)+x-1)/(sin(x)-1). - Alois P. Heinz, Apr 16 2018
MAPLE
b:= proc(u, o, t) option remember;
`if`(u+o=0, t, add(b(o+j-1, u-j, t), j=1..u)+
`if`(t=0, add(b(o-j, u-1+j, 1), j=1..o), 0))
end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..25); # Alois P. Heinz, Apr 14 2018
# second Maple program:
egf:= (cos(x)-sin(x)+x-1)/(sin(x)-1):
a:= n-> n! * coeff(series(egf, x, n+1), x, n):
seq(a(n), n=0..25); # Alois P. Heinz, Apr 16 2018
MATHEMATICA
b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, t, Sum[b[o + j - 1, u - j, t], {j, 1, u}] + If[t == 0, Sum[b[o - j, u - 1 + j, 1], {j, 1, o}], 0]];
a[n_] := b[n, 0, 0];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 19 2018, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael De Vlieger, Apr 11 2018
EXTENSIONS
a(13)-a(24) from Alois P. Heinz, Apr 14 2018
STATUS
approved