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A320980 Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of six.

%I A320980 #4 Oct 25 2018 19:22:05
%S A320980 0,1,7,514,4189,154261,1477381,44169020,493190771,13821362271,
%T A320980 177705152975,4949371839867,72355179873697,2058206624313873,
%U A320980 33818827542140211,995975339452380880,18206096557050382759,558929622195992201388,11264684856271486133087
%N A320980 Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of six.
%H A320980 Alois P. Heinz, <a href="/A320980/b320980.txt">Table of n, a(n) for n = 6..452</a>
%F A320980 a(n) = A262131(n) - A262130(n).
%p A320980 b:= proc(u, o, c) option remember; `if`(c<0 or c>6, 0, `if`(u+o=0,
%p A320980        x^c, (p-> add(coeff(p, x, i)*x^max(i, c), i=0..6))(add(
%p A320980        b(u-j, o-1+j, c-1), j=1..u)+add(b(u+j-1, o-j, c+1), j=1..o))))
%p A320980     end:
%p A320980 a:= n-> coeff(add(b(j-1, n-j, 0), j=1..n), x, 6):
%p A320980 seq(a(n), n=6..30);
%Y A320980 Column k=6 of A262125.
%Y A320980 Cf. A262130, A262131.
%K A320980 nonn
%O A320980 6,3
%A A320980 _Alois P. Heinz_, Oct 25 2018