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A262165 Number of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value <= 3.

%I A262165 #7 Aug 30 2021 06:36:43
%S A262165 1,1,2,5,19,82,454,2795,20346,162613,1469309,14424200,155842828,
%T A262165 1812646171,22807141756,306480808871,4403059520043,67100946088054,
%U A262165 1084001371054298,18469410744415367,331442882307143590,6242679740272435021,123215973021475320637
%N A262165 Number of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value <= 3.
%H A262165 Alois P. Heinz, <a href="/A262165/b262165.txt">Table of n, a(n) for n = 0..458</a>
%F A262165 a(n) = A262163(n,3).
%p A262165 b:= proc(u, o, c) option remember; `if`(c<0 or c>3, 0, `if`(u+o=0,
%p A262165        x^c, (p-> add(coeff(p, x, i)*x^max(i, c), i=0..3))(add(
%p A262165        b(u-j, o-1+j, c-1), j=1..u)+add(b(u+j-1, o-j, c+1), j=1..o))))
%p A262165     end:
%p A262165 a:= n-> (p-> add(coeff(p, x, i), i=0..min(n, 3)))(b(0, n, 0)):
%p A262165 seq(a(n), n=0..25);
%t A262165 b[u_, o_, c_] := b[u, o, c] = If[c < 0 || c > 3, 0, If[u + o == 0, x^c, Function[p, Sum[Coefficient[p, x, i]*x^Max[i, c], {i, 0, 3}]][Sum[b[u - j, o - 1 + j, c - 1], {j, u}] + Sum[b[u + j - 1, o - j, c + 1], {j, o}]]]];
%t A262165 a[n_] := Function[p, Sum[Coefficient[p, x, i], {i, 0, Min[n, 3]}]][b[0, n, 0]];
%t A262165 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Aug 30 2021, after _Alois P. Heinz_ *)
%Y A262165 Column k=3 of A262163.
%K A262165 nonn
%O A262165 0,3
%A A262165 _Alois P. Heinz_, Sep 13 2015