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

1, 0, 1, 4, 15, 52, 257, 1272, 7679, 47864, 346113, 2604380, 22022143, 194053836, 1881735169, 18998097328, 207983607807, 2366490065968, 28880901505025, 365599818496116, 4922617151619071, 68612903386404260, 1010501269355233281, 15376572385777544744

Offset: 1

Author

Emeric Deutsch, Aug 06 2009

Keywords

• nonn

Comments

a(n) = Sum_{k>=0} k*A162979(n,k).

a(2n+1) = A162977(2n+1).

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..485
• R. P. Stanley, Alternating permutations Talk slides.

Crossrefs

Cf. A000111, A162977, A162979.

Programs

Maple

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);

Mathematica

a111[n_] := If[EvenQ[n], Abs[EulerE[n]], Abs[(2^(n+1) (2^(n+1) - 1) BernoulliB[n+1])/(n+1)]];
a[n_?EvenQ] := With[{m = n/2}, a111[2m] + (-1)^m a111[0] + 2Sum[(-1)^j a111[2m - 2j], {j, 1, m-1}]];
a[n_?OddQ] := With[{m = (n-1)/2}, Sum[(-1)^j a111[2m+1-2j], {j, 0, m}]];
Array[a, 25] (* Jean-François Alcover, Jul 24 2018 *)

Formula

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.