OFFSET
0,5
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..300
FORMULA
a(n) = A291722(n,n).
a(n) ~ c * d^n / n^2, where d = 7.7572369635460295... and c = 0.022080578979754... - Vaclav Kotesovec, Nov 17 2022
EXAMPLE
a(3) = 1: 312.
a(4) = 4: 3142, 4213, 4231, 4312.
a(5) = 15: 15234, 25134, 31542, 35124, 41235, 42153, 42531, 43152, 45123, 53214, 53241, 53421, 54213, 54231, 54312.
a(6) = 104: 126354, 136254, 142635, 146253, ..., 653421, 654213, 654231, 654312.
MAPLE
b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,
add(b(u-j, o+j-1)*x^(j-1), j=1..u)+
add(b(u+j-1, o-j)*x^(j-1), j=1..o)))
end:
a:= n-> coeff(b(0, n), x, n):
seq(a(n), n=0..26);
MATHEMATICA
b[u_, o_] := b[u, o] = Expand[If[u + o == 0, 1,
Sum[b[u - j, o + j - 1]*x^(j - 1), {j, 1, u}] +
Sum[b[u + j - 1, o - j]*x^(j - 1), {j, 1, o}]]];
a[n_] := Coefficient[b[0, n], x, n];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Nov 17 2022, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 02 2017
STATUS
approved