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A302903 Number of permutations of [2n+1] having exactly n alternating descents.

%I A302903 #16 Dec 21 2020 07:17:06
%S A302903 1,2,36,1196,76840,7570716,1085246904,211595659320,53984412657360,
%T A302903 17440458896525180,6960292943873805976,3362366089440205308072,
%U A302903 1933633403768889597292336,1305355624659052356741634136,1022196734801743485304805455920,919035074839303194470918726496240
%N A302903 Number of permutations of [2n+1] having exactly n alternating descents.
%C A302903 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 A302903 Alois P. Heinz, <a href="/A302903/b302903.txt">Table of n, a(n) for n = 0..100</a>
%H A302903 D. Chebikin, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v15i1r132">Variations on descents and inversions in permutations</a>, The Electronic J. of Combinatorics, 15 (2008), #R132.
%F A302903 a(n) = A145876(2n+1,n+1) = A302905(2n+1).
%F A302903 a(n) ~ sqrt(3) * 2^(2*n + 2) * n^(2*n + 1) / (sqrt(5) * exp(2*n)). - _Vaclav Kotesovec_, Apr 29 2018
%e A302903 a(1) = 2: 123, 321.
%e A302903 a(2) = 36: 12345, 12543, 13542, 14325, 14532, 15324, 15423, 21354, 21453, 23541, 24315, 24531, 25314, 25413, 31254, 31452, 32145, 32451, 34215, 34521, 35214, 35412, 41253, 41352, 42135, 42351, 43125, 45213, 45312, 51243, 51342, 52134, 52341, 53124, 54123, 54321.
%p A302903 b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,
%p A302903        add(b(o+j-1, u-j)*x, j=1..u)+
%p A302903        add(b(o-j, u-1+j),   j=1..o)))
%p A302903     end:
%p A302903 a:= n-> coeff(b(2*n+1, 0), x, n+1):
%p A302903 seq(a(n), n=0..20);
%t A302903 b[u_, o_] := b[u, o] = Expand[If[u + o == 0, 1,
%t A302903     Sum[b[o + j - 1, u - j]*x, {j, 1, u}] +
%t A302903     Sum[b[o - j, u - 1 + j], {j, 1, o}]]];
%t A302903 a[n_] := Coefficient[b[2n + 1, 0], x, n + 1];
%t A302903 a /@ Range[0, 20] (* _Jean-François Alcover_, Dec 21 2020, after _Alois P. Heinz_ *)
%Y A302903 Bisection (odd part) of A302905.
%Y A302903 Cf. A145876.
%K A302903 nonn
%O A302903 0,2
%A A302903 _Alois P. Heinz_, Apr 15 2018