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A129817 Number of alternating fixed-point-free permutations on n letters. 4
1, 0, 1, 1, 2, 6, 24, 102, 528, 2952, 19008, 131112, 1009728, 8271792, 74167488, 703077552, 7194754368, 77437418112, 890643066048, 10726837356672, 136988469649728, 1825110309733632, 25625477737660608, 374159217291201792, 5728724202727533888, 90961591766739121152, 1508303564683904357568, 25874345243221479539712, 461932949559928514787648, 8513674175717969079785472, 162818666826944872460200128 (list; graph; refs; listen; history; text; internal format)
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 gave 4132 and 3241.

(End)

LINKS

Table of n, a(n) for n=0..30.

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

Cf. A000111, A000166, A007779.

Column k=0 of A162979.

Sequence in context: A094012 A141253 A078486 * A230797 A128652 A152316

Adjacent sequences:  A129814 A129815 A129816 * A129818 A129819 A129820

KEYWORD

more,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

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Last modified October 21 12:52 EDT 2018. Contains 316422 sequences. (Running on oeis4.)