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%I #28 Jan 28 2022 06:26:54
%S 1,0,2,2,16,64,416,2848,22912,205952,2060032,22659328,271913984,
%T 3534877696,49488295936,742324422656,11877190795264,201912243453952,
%U 3634420382302208,69053987263479808,1381079745270120448,29002674650671480832,638058842314774675456
%N Number of permutations of n elements with an even number of fixed points.
%C Let d(n) be the number of derangements of n elements (sequence A000166) then a(n) has the recursion: a(n) = d(n) + C(n,2)*d(n-2) + C(n,4)*d(n-4) + C(n,6)*d(n-6)... = A000166(n) + A000387(n) + A000475(n) + C(n,6)*d(n-6)... The E.g.f. for a(n) is: cosh(x) * exp(-x)/(1-x) and the asymptotic expression for a(n) is: a(n) ~ n! * (1 + 1/e^2)/2 i.e., as n goes to infinity the fraction of permutations that has an even number of fixed points is about (1 + 1/e^2)/2 = 0.567667...
%H G. C. Greubel, <a href="/A062282/b062282.txt">Table of n, a(n) for n = 0..445</a>
%F a(n) = Sum_{k=0..[n/2]} Sum_{l=0..(n-2*k)} (-1)^l * n!/((2*k)! * l!).
%F More generally, e.g.f. for number of degree-n permutations with an even number of k-cycles is cosh(x^k/k)*exp(-x^k/k)/(1-x). - _Vladeta Jovovic_, Jan 31 2006
%F E.g.f.: 1/(1-x)/(x*E(0)+1), where E(k) = 1 - x^2/( x^2 + (2*k+1)*(2*k+3)/E(k+1) ); (continued fraction ). - _Sergei N. Gladkovskii_, Dec 29 2013
%F Conjecture: a(n) = Sum_{k=0..n} A008290(n, k)*A059841(k). - _John Keith_, Jun 30 2020
%t nn = 20; d = Exp[-x]/(1 - x); Range[0, nn]! CoefficientList[Series[Cosh[x] d, {x, 0, nn}], x] (* _Geoffrey Critzer_, Jan 14 2012 *)
%t Table[Sum[Sum[(-1)^j * n!/(j!*(2*k)!), {j, 0, n - 2*k}], {k, 0, Floor[n/2]}], {n,0,50}] (* _G. C. Greubel_, Aug 21 2017 *)
%o (PARI) for(n=0,50, print1(sum(k=0,n\2, sum(j=0,n-2*k, (-1)^j*n!/(j!*(2*k)!))), ", ")) \\ _G. C. Greubel_, Aug 21 2017
%Y Cf. A000166, A000387, A000475.
%Y Cf. A063083, A100818, A092295, A111752, A111753, A111723, A111724, A088336, A088506.
%K nonn
%O 0,3
%A Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 04 2001
%E More terms from _Vladeta Jovovic_, Jul 05 2001
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