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A186770 Number of permutations of {1,2,...,n} having no nonincreasing even cycles. A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1) . A cycle is said to be even if it has an even number of entries.

%I A186770 #11 May 18 2017 10:14:26
%S A186770 1,1,2,6,19,95,451,3157,21092,189828,1660351,18263861,197541565,
%T A186770 2568040345,33029787974,495446819610,7377279473779,125413751054243,
%U A186770 2120559951767503,40290639083582557,762353357154540584,16009420500245352264
%N A186770 Number of permutations of {1,2,...,n} having no nonincreasing even cycles. A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)<b(2)<b(3)<... .  A cycle is said to be even if it has an even number of entries.
%C A186770 a(n) = A186769(n,0).
%H A186770 Alois P. Heinz, <a href="/A186770/b186770.txt">Table of n, a(n) for n = 0..450</a>
%F A186770 E.g.f.: g(z) = exp(cosh z - 1)*sqrt((1+z)/(1-z)).
%e A186770 a(4)=19 because among the 24 permutations of {1,2,3,4} only (1243), (1324), (1342), (1423), and (1432) have nonincreasing even cycles.
%p A186770 g := exp(cosh(z)-1)*sqrt((1+z)/(1-z)): gser := series(g, z = 0, 27): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 21);
%p A186770 # second Maple program:
%p A186770 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
%p A186770       binomial(n-1, j-1)*`if`(j::odd, (j-1)!, 1), j=1..n))
%p A186770     end:
%p A186770 seq(a(n), n=0..25);  # _Alois P. Heinz_, Apr 13 2017
%t A186770 a[n_] := a[n] = If[n==0, 1, Sum[a[n-j]*Binomial[n-1, j-1]*If[OddQ[j], (j-1)!, 1], {j, 1, n}]];
%t A186770 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, May 18 2017, after _Alois P. Heinz_ *)
%Y A186770 Cf. A186761, A186762, A186764, A186765, A186766, A186767, A186769.
%K A186770 nonn
%O A186770 0,3
%A A186770 _Emeric Deutsch_, Feb 27 2011