This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A288912 #12 Aug 31 2021 07:41:45
%S A288912 1,1,2,6,23,106,558,3284,21200,148539,1119273,9013112,77106652,
%T A288912 697811164,6652604804,66593158893,697756930786,7633155361594,
%U A288912 86969814549075,1029939820075074,12652809025029242,160977119684852369,2117642963178349336,28763717105362639324
%N A288912 Number of permutations p of [n] such that 0p has a nonincreasing down-jump sequence.
%C A288912 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 A288912 Alois P. Heinz, <a href="/A288912/b288912.txt">Table of n, a(n) for n = 0..200</a>
%p A288912 b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
%p A288912 add(b(u-j, o+j-1, j), j=1..min(t, u))+
%p A288912 add(b(u+j-1, o-j, t), j=1..o))
%p A288912 end:
%p A288912 a:= n-> b(0, n$2):
%p A288912 seq(a(n), n=0..30);
%t A288912 b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1,
%t A288912 Sum[b[u - j, o + j - 1, j], {j, Min[t, u]}] +
%t A288912 Sum[b[u + j - 1, o - j, t], {j, o}]];
%t A288912 a[n_] := b[0, n, n];
%t A288912 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Aug 31 2021, after _Alois P. Heinz_ *)
%Y A288912 Cf. A288910, A288911, A291685.
%K A288912 nonn
%O A288912 0,3
%A A288912 _Alois P. Heinz_, Sep 01 2017