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A097597 Number of permutations of [n] with no increasing runs of even length.

%I A097597 #34 Nov 30 2018 17:17:58
%S A097597 1,1,1,2,7,25,102,531,3141,20218,146215,1174889,10225678,96226363,
%T A097597 978420285,10657592850,123672458583,1525420453945,19929519469558,
%U A097597 274771355003651,3987385414116085,60764250319690666,970085750385722631,16190361659675002857
%N A097597 Number of permutations of [n] with no increasing runs of even length.
%H A097597 Alois P. Heinz, <a href="/A097597/b097597.txt">Table of n, a(n) for n = 0..474</a>
%H A097597 Ira M. Gessel, <a href="http://people.brandeis.edu/~gessel/homepage/papers/thesis.pdf">Generating Functions and Enumeration of Sequences</a>, Ph.D. thesis, MIT, 1977, p. 52.
%H A097597 Toufik Mansour and Mark Shattuck, <a href="https://www.researchgate.net/publication/230799578_A_combinatorial_proof_of_a_result_for_permutation_pairs/download">A combinatorial proof of a result for permutation pairs</a>, Central European Journal of Mathematics, 10 (2012), 797-806.
%F A097597 E.g.f.: sqrt(5)/(sqrt(5)-2*exp(-x/2)*sinh(sqrt(5)*x/2)).
%F A097597 E.g.f.: (1 + Sum_{n>=1} (-1)^n F_n x^n/n!)^(-1), where F_n is the n-th Fibonacci number. - _Ira M. Gessel_, Jul 27 2014
%F A097597 a(n) ~ n! * sinh(r*sqrt(5)) / (2^n*r^(n+1)*(sqrt(5)*cosh(r*sqrt(5))-sinh(r*sqrt(5)))), where r = 0.68903745689226... is the root of the equation 1-exp(-2*sqrt(5)*r) = sqrt(5)*exp((1-sqrt(5))*r). - _Vaclav Kotesovec_, Sep 29 2013
%e A097597 a(4) = 7 because 2/134, 3/124, 4/123, 234/1, 134/2, 124/3 and 4/3/2/1 are the only permutations of [4] with no increasing runs of even length.
%p A097597 G:=sqrt(5)/(sqrt(5)-2*exp(-x/2)*sinh(sqrt(5)*x/2)): Gser:=simplify(series(G,x=0,25)): 1,seq(n!*coeff(Gser,x^n),n=1..24);
%p A097597 # second Maple program:
%p A097597 b:= proc(u, o, t) option remember; `if`(u+o=0, t,
%p A097597       add(b(u+j-1, o-j, irem(t+1, 2)), j=1..o)+
%p A097597       `if`(t=0, 0, add(b(u-j, o+j-1, 1), j=1..u)))
%p A097597     end:
%p A097597 a:= n-> b(n, 0, 1):
%p A097597 seq(a(n), n=0..25);  # _Alois P. Heinz_, Nov 19 2013
%t A097597 CoefficientList[Series[Sqrt[5]/(Sqrt[5]-2*E^(-x/2)*(E^(Sqrt[5]*x/2)/2 - E^(-Sqrt[5]*x/2)/2)), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 29 2013 *)
%Y A097597 Column k=0 of A097592.
%Y A097597 Cf. A000045.
%K A097597 nonn
%O A097597 0,4
%A A097597 _Emeric Deutsch_, Aug 29 2004