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A291688 Number of permutations p of [2n] such that 0p has a nonincreasing jump sequence beginning with n.

%I A291688 #13 Aug 30 2021 09:39:33
%S A291688 1,1,5,36,327,3392,38795,469662,5935728,77416352,1035050705,
%T A291688 14094000938,195075365778,2734475097609,38747262233793,
%U A291688 554199475506095,7990492729051526,115995691148658656,1694340616136589743,24882428969673439384,367160435328847044586
%N A291688 Number of permutations p of [2n] such that 0p has a nonincreasing jump sequence beginning with n.
%C A291688 An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here.
%H A291688 Alois P. Heinz, <a href="/A291688/b291688.txt">Table of n, a(n) for n = 0..250</a>
%F A291688 a(n) = A291684(2n,n).
%e A291688 a(2) = 5: 2134, 2314, 2341, 2413, 2431.
%p A291688 b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
%p A291688       add(b(u-j, o+j-1, j), j=1..min(t, u))+
%p A291688       add(b(u+j-1, o-j, j), j=1..min(t, o)))
%p A291688     end:
%p A291688 a:= n-> b(0, 2*n, n)-`if`(n=0, 0, b(0, 2*n, n-1)):
%p A291688 seq(a(n), n=0..25);
%t A291688 b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1,
%t A291688      Sum[b[u - j, o + j - 1, j], {j, Min[t, u]}] +
%t A291688      Sum[b[u + j - 1, o - j, j], {j, Min[t, o]}]];
%t A291688 a[n_] :=  b[0, 2n, n] - If[n == 0, 0, b[0, 2n, n - 1]];
%t A291688 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Aug 30 2021, after _Alois P. Heinz_ *)
%Y A291688 Cf. A291684.
%Y A291688 Bisection (even part) of A303204.
%K A291688 nonn
%O A291688 0,3
%A A291688 _Alois P. Heinz_, Aug 29 2017