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A320977 Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of three.

%I A320977 #5 Oct 25 2018 19:10:48
%S A320977 0,1,4,57,311,3857,27589,355751,3185258,44435241,479927081,7311147055,
%T A320977 92603527295,1540826943617,22398016781688,405865410503155,
%U A320977 6658173453910401,130871330990044897,2390845313044048301,50759828657781860167,1021623022236754343662
%N A320977 Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of three.
%H A320977 Alois P. Heinz, <a href="/A320977/b320977.txt">Table of n, a(n) for n = 3..458</a>
%F A320977 a(n) = A262128(n) - A262126(n).
%p A320977 b:= proc(u, o, c) option remember; `if`(c<0 or c>3, 0, `if`(u+o=0,
%p A320977        x^c, (p-> add(coeff(p, x, i)*x^max(i, c), i=0..3))(add(
%p A320977        b(u-j, o-1+j, c-1), j=1..u)+add(b(u+j-1, o-j, c+1), j=1..o))))
%p A320977     end:
%p A320977 a:= n-> coeff(add(b(j-1, n-j, 0), j=1..n), x, 3):
%p A320977 seq(a(n), n=3..30);
%Y A320977 Column k=3 of A262125.
%Y A320977 Cf. A262126, A262128.
%K A320977 nonn
%O A320977 3,3
%A A320977 _Alois P. Heinz_, Oct 25 2018