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A320978 Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of four.

%I A320978 #4 Oct 25 2018 19:15:10
%S A320978 0,1,5,123,778,14126,111811,1957924,18846572,342736488,3904186779,
%T A320978 75626480215,996651530998,20739117286767,310506509296055,
%U A320978 6952893022274589,116561453850733664,2805243168612301480,52046467944203248643,1342921799608903158968
%N A320978 Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of four.
%H A320978 Alois P. Heinz, <a href="/A320978/b320978.txt">Table of n, a(n) for n = 4..455</a>
%F A320978 a(n) = A262129(n) - A262128(n).
%p A320978 b:= proc(u, o, c) option remember; `if`(c<0 or c>4, 0, `if`(u+o=0,
%p A320978        x^c, (p-> add(coeff(p, x, i)*x^max(i, c), i=0..4))(add(
%p A320978        b(u-j, o-1+j, c-1), j=1..u)+add(b(u+j-1, o-j, c+1), j=1..o))))
%p A320978     end:
%p A320978 a:= n-> coeff(add(b(j-1, n-j, 0), j=1..n), x, 4):
%p A320978 seq(a(n), n=4..30);
%Y A320978 Column k=4 of A262125.
%Y A320978 Cf. A262128, A262129.
%K A320978 nonn
%O A320978 4,3
%A A320978 _Alois P. Heinz_, Oct 25 2018