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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.

%I A262167 #7 Aug 30 2021 06:37:35
%S A262167 1,1,2,5,20,87,521,3264,25899,212534,2092218,21250451,249294149,
%T A262167 3018733862,41077515364,577524896681,8940290166542,143098583946093,
%U A262167 2483312451690110,44571301924473611,857112705946351481,17044616630699383294,359813788663496645489
%N 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.
%H A262167 Alois P. Heinz, <a href="/A262167/b262167.txt">Table of n, a(n) for n = 0..453</a>
%F A262167 a(n) = A262163(n,5).
%p A262167 b:= proc(u, o, c) option remember; `if`(c<0 or c>5, 0, `if`(u+o=0,
%p A262167        x^c, (p-> add(coeff(p, x, i)*x^max(i, c), i=0..5))(add(
%p A262167        b(u-j, o-1+j, c-1), j=1..u)+add(b(u+j-1, o-j, c+1), j=1..o))))
%p A262167     end:
%p A262167 a:= n-> (p-> add(coeff(p, x, i), i=0..min(n, 5)))(b(0, n, 0)):
%p A262167 seq(a(n), n=0..25);
%t A262167 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}]]]];
%t A262167 a[n_] := Function[p, Sum[Coefficient[p, x, i], {i, 0, Min[n, 5]}]][b[0, n, 0]];
%t A262167 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Aug 30 2021, after _Alois P. Heinz_ *)
%Y A262167 Column k=5 of A262163.
%K A262167 nonn
%O A262167 0,3
%A A262167 _Alois P. Heinz_, Sep 13 2015