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A316390 Number of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value of three.

%I A316390 #7 Jul 01 2018 13:24:20
%S A316390 1,3,28,130,1263,8090,88101,724189,8887448,89401804,1229179691,
%T A316390 14638611036,223711095367,3078744103979,51892788554614,
%U A316390 810254535452378,14955918856848519,261173044555806630,5245841953983851853,101285541723126490941,2201267668629421856324
%N A316390 Number of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value of three.
%H A316390 Alois P. Heinz, <a href="/A316390/b316390.txt">Table of n, a(n) for n = 3..458</a>
%F A316390 a(n) = A262165(n) - A262164(n).
%p A316390 b:= proc(u, o, c, k) option remember;
%p A316390       `if`(c<0 or c>k, 0, `if`(u+o=0, 1,
%p A316390        add(b(u-j, o-1+j, c+1, k), j=1..u)+
%p A316390        add(b(u+j-1, o-j, c-1, k), j=1..o)))
%p A316390     end:
%p A316390 a:= n-> b(n, 0$2, 3)-b(n, 0$2, 2):
%p A316390 seq(a(n), n=3..23);
%Y A316390 Column k=3 of A258829.
%Y A316390 Cf. A262164, A262165.
%K A316390 nonn
%O A316390 3,2
%A A316390 _Alois P. Heinz_, Jul 01 2018