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A262167
Number of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value <= 5.
4
1, 1, 2, 5, 20, 87, 521, 3264, 25899, 212534, 2092218, 21250451, 249294149, 3018733862, 41077515364, 577524896681, 8940290166542, 143098583946093, 2483312451690110, 44571301924473611, 857112705946351481, 17044616630699383294, 359813788663496645489
OFFSET
0,3
LINKS
FORMULA
a(n) = A262163(n,5).
MAPLE
b:= proc(u, o, c) option remember; `if`(c<0 or c>5, 0, `if`(u+o=0,
x^c, (p-> add(coeff(p, x, i)*x^max(i, c), i=0..5))(add(
b(u-j, o-1+j, c-1), j=1..u)+add(b(u+j-1, o-j, c+1), j=1..o))))
end:
a:= n-> (p-> add(coeff(p, x, i), i=0..min(n, 5)))(b(0, n, 0)):
seq(a(n), n=0..25);
MATHEMATICA
b[u_, o_, c_] := b[u, o, c] = If[c < 0 || c > 5, 0, If[u + o == 0, x^c, Function[p, Sum[Coefficient[p, x, i]*x^Max[i, c], {i, 0, 5}]][Sum[b[u - j, o - 1 + j, c - 1], {j, u}] + Sum[b[u + j - 1, o - j, c + 1], {j, o}]]]];
a[n_] := Function[p, Sum[Coefficient[p, x, i], {i, 0, Min[n, 5]}]][b[0, n, 0]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 30 2021, after Alois P. Heinz *)
CROSSREFS
Column k=5 of A262163.
Sequence in context: A363140 A190656 A262166 * A262168 A262169 A262170
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 13 2015
STATUS
approved