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%I #30 Nov 24 2021 03:04:39
%S 1,71,45541,120686411,908138776681,15611712012050351,
%T 531909061958526321421,32491881630252866646683891,
%U 3302814916156503291298772711761,527393971346575736206847604137659031,126355819963625435928020023737689391659701
%N 2 up, 2 down, ..., 2 up, 2 down, 2 up permutations of length 4n+3.
%C Bisection of A005981. The entries listed above suggest various congruences for a(n): a(n) = 1 (mod 10), a(n) = 1 + 70*n (mod 100), a(n) = 1 + 70*n + 200*n(n-1) (mod 1000). Are these congruences true for all n? For an arbitrary integer m, the sequence a(n) taken modulo m may eventually become periodic. Compare with A081727.
%H Alois P. Heinz, <a href="/A131454/b131454.txt">Table of n, a(n) for n = 0..50</a>
%H Christopher R. H. Hanusa, Alejandro H. Morales, and Martha Yip, <a href="https://arxiv.org/abs/2107.07326">Column convex matrices, G-cyclic orders, and flow polytopes</a>, arXiv:2107.07326 [math.CO], 2021.
%H 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 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 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 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 E.g.f.: Sum_{n>=0} a(n)*(x^(4n+3))/(4n+3)! = (exp(2x)-2*sin(x)*exp(x)-1)/(2*exp(x)+cos(x)*(exp(2x)+1)). It appears that a(n) = (4n+3)!*coefficient of x^(4n+3) in the Taylor expansion of -4/(2*exp(x)+cos(x)*(exp(2x)+1)).
%e (1 4 5 3 2 6 7) is a 2 up, 2 down, 2 up permutation - one of the seventy-one permutations of this type in the symmetric group on 7 letters.
%p g:=(sinh(x)-sin(x))/(cos(x)*cosh(x)+1): series(%,x,44):
%p seq(n!*coeff(%,x,n),n=3..45,4); # _Peter Luschny_, Feb 07 2017
%t Table[(CoefficientList[Series[(-Sin[x] + Sinh[x]) / (1 + Cos[x]*Cosh[x]), {x, 0, 60}], x] * Range[0, 59]!)[[n]], {n, 4, 60, 4}] (* _Vaclav Kotesovec_, Sep 09 2014 *)
%Y Cf. A000111, A005981, A058257, A131453, A131455.
%K easy,nonn
%O 0,2
%A _Peter Bala_, Jul 13 2007
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