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A288911 Number of permutations p of [n] such that 0p has a nonincreasing up-jump sequence.

%I A288911 #13 Aug 30 2021 06:39:54
%S A288911 1,1,2,5,19,80,416,2306,14588,98053,724183,5633793,47416901,417050215,
%T A288911 3914152702,38288228393,395496623939,4241350801439,47715403637219,
%U A288911 555476398869869,6744406721447538,84548532634924758,1100301545470162305,14751287346427752887
%N A288911 Number of permutations p of [n] such that 0p has a nonincreasing up-jump sequence.
%C A288911 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 A288911 Alois P. Heinz, <a href="/A288911/b288911.txt">Table of n, a(n) for n = 0..200</a>
%p A288911 b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
%p A288911       add(b(u-j, o+j-1, t), j=1..u)+
%p A288911       add(b(u+j-1, o-j, j), j=1..min(t, o)))
%p A288911     end:
%p A288911 a:= n-> b(0, n$2):
%p A288911 seq(a(n), n=0..30);
%t A288911 b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1,
%t A288911      Sum[b[u - j, o + j - 1, t], {j, u}] +
%t A288911      Sum[b[u + j - 1, o - j, j], {j, Min[t, o]}]];
%t A288911 a[n_] := b[0, n, n];
%t A288911 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Aug 30 2021, after _Alois P. Heinz_ *)
%Y A288911 Cf. A288910, A288912, A291685.
%K A288911 nonn
%O A288911 0,3
%A A288911 _Alois P. Heinz_, Sep 01 2017