%I #18 Dec 21 2020 11:55:57
%S 1,1,7,182,8699,704834,84889638,14322115212,3216832016019,
%T 928559550102410,334876925319944690,147563833511292247796,
%U 78009671642511668089822,48728981875112003682759892,35506576774281843111748649644,29848802048200930275501944893080
%N Number of permutations of [2n] having exactly n-1 alternating descents.
%C a(0) = 1 by convention.
%C Index i is an alternating descent of permutation p if either i is odd and p(i) > p(i+1), or i is even and p(i) < p(i+1).
%H Alois P. Heinz, <a href="/A302904/b302904.txt">Table of n, a(n) for n = 0..100</a>
%H D. Chebikin, <a href="https://doi.org/10.37236/856">Variations on descents and inversions in permutations</a>, The Electronic J. of Combinatorics, 15 (2008), #R132.
%F a(n) = A145876(2n,n) = A302905(2n) for n > 0.
%F a(n) ~ sqrt(3) * 2^(2*n + 1) * n^(2*n) / (sqrt(5) * exp(2*n)). - _Vaclav Kotesovec_, Apr 29 2018
%e a(2) = 7: 1234, 1432, 2431, 3214, 3421, 4213, 4312.
%p b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,
%p add(b(o+j-1, u-j)*x, j=1..u)+
%p add(b(o-j, u-1+j), j=1..o)))
%p end:
%p a:= n-> coeff(b(2*n, 0), x, n):
%p seq(a(n), n=0..20);
%t b[u_, o_] := b[u, o] = Expand[If[u + o == 0, 1,
%t Sum[b[o + j - 1, u - j] x, {j, 1, u}] +
%t Sum[b[o - j, u - 1 + j], {j, 1, o}]]];
%t a[n_] := Coefficient[b[2 n, 0], x, n];
%t a /@ Range[0, 20] (* _Jean-François Alcover_, Dec 21 2020, after _Alois P. Heinz_ *)
%Y Bisection (even part) of A302905.
%Y Cf. A145876.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Apr 15 2018