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A303203 Number of permutations p of [2n+1] such that 0p has a nonincreasing jump sequence beginning with n+1.

%I A303203 #14 Sep 01 2021 03:40:29
%S A303203 1,2,12,98,988,10872,129520,1609176,20786804,274792342,3719296036,
%T A303203 51119572738,712918642042,10055531355652,143287150725298,
%U A303203 2058368140071146,29796314629826814,434051906728752164,6359063811707227298,93635249115751389952,1385028309353547034876
%N A303203 Number of permutations p of [2n+1] such that 0p has a nonincreasing jump sequence beginning with n+1.
%C A303203 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 A303203 Alois P. Heinz, <a href="/A303203/b303203.txt">Table of n, a(n) for n = 0..250</a>
%F A303203 a(n) = A291684(2n+1,n+1).
%p A303203 b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
%p A303203       add(b(sort([u-j, o+j-1])[], j), j=1..min(t, u))+
%p A303203       add(b(sort([u+j-1, o-j])[], j), j=1..min(t, o)))
%p A303203     end:
%p A303203 a:= n-> b(0, 2*n+1, n+1)-b(0, 2*n+1, n):
%p A303203 seq(a(n), n=0..25);
%t A303203 b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1,
%t A303203      Sum[b[u-j, o+j-1, j], {j, 1, Min[t, u]}] +
%t A303203      Sum[b[u+j-1, o-j, j], {j, 1, Min[t, o]}]];
%t A303203 a[n_] := b[0, 2n+1, n+1] - b[0, 2n+1, n];
%t A303203 a /@ Range[0, 25] (* _Jean-François Alcover_, Sep 01 2021, after _Alois P. Heinz_ *)
%Y A303203 Bisection (odd part) of A303204.
%Y A303203 Cf. A291684.
%K A303203 nonn
%O A303203 0,2
%A A303203 _Alois P. Heinz_, Apr 19 2018