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A246015 Number of permutations of n letters that have all valleys even and all peaks odd and that end with an ascent. 1
0, 1, 1, 3, 4, 34, 70, 853, 2365, 36369, 125870, 2367212, 9829820, 218570717, 1058927507, 27167018427, 150421002736, 4373595961678, 27243141998506, 885304614118777, 6127294825424317, 220074660203753961, 1675478215412320154, 65909583942401516528, 547401364007816870552, 23406034074310130665241 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..470 (first 101 terms from N. J. A. Sloane)
Ira M. Gessel and Yan Zhuang, Counting permutations by alternating descents, arXiv:1408.1886 [math.CO], 2014. See g(n).
FORMULA
See Maple code for recurrence involving A246012(n) = f(n).
MAPLE
Gf1:= (3*sin(x/2)+3*cosh(sqrt(3)*x/2))/(3*cos(x/2)-sqrt(3)*sinh(sqrt(3)*x/2));
t1:=series(Gf1, x, 40):
t2:=SERIESTOLISTMULT(t1);
f:=n->t2[n+1]; # this is f(n), A246012
g:=proc(n) global f; option remember; local m, s, k; m:=n-1;
if n=0 then 0 else
s:=add(binomial(m, 2*k)*f(2*k)*(f(m-2*k)-g(m-2*k)), k=0..floor((m-2)/2));
if m mod 2 = 0 then s+f(m) else s+g(m); fi;
fi; end;
[seq(g(n), n=0..30)];
# second Maple program:
b:= proc(u, o, p, t) option remember; `if`(u+o=0, t,
`if`(t=0 or p=1, add(b(u-j, o+j-1, 1-p, 0), j=1..u), 0)+
`if`(t=1 or p=0, add(b(u+j-1, o-j, 1-p, 1), j=1..o), 0))
end:
a:= n-> b(0, n, 0$2):
seq(a(n), n=0..25); # Alois P. Heinz, Aug 03 2018
MATHEMATICA
f[n_] := SeriesCoefficient[(3*Cosh[(Sqrt[3]*x)/2] + 3*Sin[x/2])/(3*Cos[x/2] - Sqrt[3]*Sinh[(Sqrt[3]*x)/2]), {x, 0, n}]*n!;
g[n_] := g[n] = Module[{m, s, k}, m = n-1; If[n == 0, 0, s = Sum[ Binomial[ m, 2*k]*f[2*k]*(f[m-2*k] - g[m-2*k]), {k, 0, Floor[(m-2)/2]}]; If[EvenQ[m], s + f[m], s + g[m]]]];
Table[g[n], {n, 0, 30}] (* Jean-François Alcover, Jan 23 2018, translated from Maple *)
CROSSREFS
Cf. A246012.
Sequence in context: A296250 A032834 A025140 * A078504 A066090 A251259
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Aug 11 2014
STATUS
approved

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Last modified April 14 20:39 EDT 2024. Contains 371667 sequences. (Running on oeis4.)