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A162978 Number of fixed points in all alternating (i.e., down-up) permutations of {1,2,...,n}.

%I A162978 #16 Jul 24 2018 02:57:40
%S A162978 1,0,1,4,15,52,257,1272,7679,47864,346113,2604380,22022143,194053836,
%T A162978 1881735169,18998097328,207983607807,2366490065968,28880901505025,
%U A162978 365599818496116,4922617151619071,68612903386404260,1010501269355233281,15376572385777544744
%N A162978 Number of fixed points in all alternating (i.e., down-up) permutations of {1,2,...,n}.
%C A162978 a(n) = Sum_{k>=0} k*A162979(n,k).
%C A162978 a(2n+1) = A162977(2n+1).
%H A162978 Alois P. Heinz, <a href="/A162978/b162978.txt">Table of n, a(n) for n = 1..485</a>
%H A162978 R. P. Stanley, <a href="http://math.mit.edu/~rstan/transparencies/ida.pdf">Alternating permutations</a> Talk slides.
%F A162978 a(2n) = E(2n) + (-1)^n*E(0) + 2*Sum_{j=1..n-1} (-1)^j*E(2n-2j), a(2n+1) = Sum_{j=0..n} (-1)^j*E(2n+1-2j), where E(i) = A000111(i) are the Euler (or up-down) numbers.
%e A162978 a(4)=4 because in the 5 (=A000111(4)) down-up permutations of {1,2,3,4}, namely 4132, 3142, 2143, 4231, and 3241, we have a total of 1+0+0+2+1=4 fixed points.
%p A162978 E := sec(x)+tan(x): Eser := series(E, x = 0, 30): for n from 0 to 27 do E[n] := factorial(n)*coeff(Eser, x, n) end do: for n to 12 do a[2*n] := E[2*n]+(-1)^n*E[0]+2*add((-1)^j*E[2*n-2*j], j = 1 .. n-1) end do: for n from 0 to 12 do a[2*n+1] := add((-1)^j*E[2*n+1-2*j], j = 0 .. n) end do: seq(a[n], n = 1 .. 25);
%t A162978 a111[n_] := If[EvenQ[n], Abs[EulerE[n]], Abs[(2^(n+1) (2^(n+1) - 1) BernoulliB[n+1])/(n+1)]];
%t A162978 a[n_?EvenQ] := With[{m = n/2}, a111[2m] + (-1)^m a111[0] + 2Sum[(-1)^j a111[2m - 2j], {j, 1, m-1}]];
%t A162978 a[n_?OddQ] := With[{m = (n-1)/2}, Sum[(-1)^j a111[2m+1-2j], {j, 0, m}]];
%t A162978 Array[a, 25] (* _Jean-François Alcover_, Jul 24 2018 *)
%Y A162978 Cf. A000111, A162977, A162979.
%K A162978 nonn
%O A162978 1,4
%A A162978 _Emeric Deutsch_, Aug 06 2009