OFFSET
0,5
COMMENTS
For n > 0, a(2n-1) = A129815(2n-1); for n > 1, a(2n) = A129815(2n) + A129815(2n-2). - Vladimir Shevelev, Apr 29 2008
We conjecture that for n >= 3, A000111(2n)/a(2n) < e < A000111(2n)/A129815(2n), so that A000111(2n)/a(2n) increases while A000111(2n)/A129815(2n) decreases (and both quotients tend to e). - Vladimir Shevelev, Apr 29 2008
From Emeric Deutsch, Aug 06 2009: (Start)
Alternating permutations are also called down-up permutations.
a(n) is also the number of alternating permutations of {1,2,...,n} having exactly 1 fixed point (see the Richard Stanley reference). Example: a(4)=2 because we have 4132 and 3241.
(End)
LINKS
R. P. Stanley, Alternating permutations and symmetric functions, arXiv:math/0603520 [math.CO], 2006.
FORMULA
a(n) = A162979(n,0). - Alois P. Heinz, Nov 24 2017
EXAMPLE
a(4) = 2 because we have 3142 and 2143. - Emeric Deutsch, Aug 06 2009
MATHEMATICA
nmax = 30;
fo = Exp[e*(ArcTan[q*t] - ArcTan[t])]/(1 - e*t);
fe = Sqrt[(1+t^2)/(1+q^2*t^2)]*Exp[e*(ArcTan[q*t] - ArcTan[t])]/(1-e*t);
Q[n_] := If [OddQ[n] , SeriesCoefficient[fo, {t, 0, n}], SeriesCoefficient[fe, {t, 0, n}]] // Expand;
b[n_] := n!*SeriesCoefficient[Sec[x] + Tan[x], {x, 0, n}];
P[n_] := (Q[n] /. e^k_Integer :> b[k]) /. e :> b[1] // Expand;
a[n_] := Coefficient[P[n], q, 0];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, nmax}] (* Jean-François Alcover, Jul 24 2018 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, May 20 2007
EXTENSIONS
a(21) from Alois P. Heinz, Nov 06 2015
a(0)=1 prepended by Alois P. Heinz, Nov 24 2017
a(22)..a(30) from Jean-François Alcover, Jul 24 2018
STATUS
approved