OFFSET
0,2
COMMENTS
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.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..250
FORMULA
a(n) = A291684(2n+1,n+1).
MAPLE
b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
add(b(sort([u-j, o+j-1])[], j), j=1..min(t, u))+
add(b(sort([u+j-1, o-j])[], j), j=1..min(t, o)))
end:
a:= n-> b(0, 2*n+1, n+1)-b(0, 2*n+1, n):
seq(a(n), n=0..25);
MATHEMATICA
b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1,
Sum[b[u-j, o+j-1, j], {j, 1, Min[t, u]}] +
Sum[b[u+j-1, o-j, j], {j, 1, Min[t, o]}]];
a[n_] := b[0, 2n+1, n+1] - b[0, 2n+1, n];
a /@ Range[0, 25] (* Jean-François Alcover, Sep 01 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Apr 19 2018
STATUS
approved