cp's OEIS Frontend

A131453 2 up, 2 down, ..., 2 up, 2 down permutations of length 4n+1.

%I A131453 #27 Aug 13 2019 04:02:01
%S A131453 1,6,1456,2020656,9336345856,108480272749056,2664103110372192256,
%T A131453 122840808510269863827456,9758611490955498257378246656,
%U A131453 1251231616578606273788469919481856,245996119743058288132230759497577005056,71155698830255977656506481145458378597728256
%N A131453 2 up, 2 down, ..., 2 up, 2 down permutations of length 4n+1.
%C A131453 Bisection of A005981.
%H A131453 Alois P. Heinz, <a href="/A131453/b131453.txt">Table of n, a(n) for n = 0..50</a>
%H A131453 L. Olivier, <a href="https://gdz.sub.uni-goettingen.de/id/PPN243919689_0002?tify=%7B%22pages%22:[257]%7D">Bemerkungen über eine Art von Functionen, welche ähnliche Eigenschaften haben, wie der Cosinus und Sinus</a>, J. Reine Angew. Math. 2 (1827), 243-251.
%H A131453 H. Prodinger and T. A. Tshifhumulo, <a href="https://doi.org/10.1007/PL00012585">On q-Olivier Functions</a>, Annals of Combinatorics 6 (2002), 181-194.
%H A131453 B. Shapiro and A. Vainshtein, <a href="https://arxiv.org/abs/math/0209062">Counting real rational functions with all real critical values</a>, arXiv:math/0209062 [math.AG], 2002.
%H A131453 B. Shapiro and A. Vainshtein, <a href="http://www.ams.org/distribution/mmj/vol3-2-2003/shapiro-vainshtein.pdf">Counting real rational functions with all real critical values</a>, Moscow Math. J. 3 (2003), 647-659.
%F A131453 E.g.f.: Sum_{n>=0} a(n)*(x^(4n+1))/(4n+1)! = (sin(x)*(exp(2x)+1)+cos(x)*(exp(2x)-1))/(2*exp(x)+cos(x)*(exp(2x)+1)).
%e A131453 a(1) = 6. The six 2 up, 2 down permutations on 5 letters are (12543), (13542), (14532), (23541), (24532) and (34521).
%p A131453 g:= (tan(x)+exp(2*x)*(tan(x)+1)-1)/(exp(2*x)+2*exp(x)*sec(x)+1): series(%,x,46):
%p A131453 seq(n!*coeff(%,x,n), n=1..45,4); # _Peter Luschny_, Feb 07 2017
%t A131453 Table[(CoefficientList[Series[((-1 + E^(2*x))*Cos[x] + (1 + E^(2*x))*Sin[x]) / (2*E^x + (1 + E^(2*x))* Cos[x]), {x, 0, 80}], x] * Range[0, 77]!)[[n]], {n, 2, 78, 4}] (* _Vaclav Kotesovec_, Sep 09 2014 *)
%Y A131453 Cf. A000111, A005981, A058257, A131454, A131455.
%K A131453 easy,nonn
%O A131453 0,2
%A A131453 _Peter Bala_, Jul 13 2007