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A320983 Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of nine.

%I A320983 #4 Oct 25 2018 19:32:51
%S A320983 0,1,10,4116,45431,4619654,56724741,3604309466,49361628592,
%T A320983 2539183745028,38672184949164,1795814647499288,30258670886635123,
%U A320983 1339822976545126729,24829628782352186543,1080751193228180109916,21899879370157544611142,954090805109110152360526
%N A320983 Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of nine.
%H A320983 Alois P. Heinz, <a href="/A320983/b320983.txt">Table of n, a(n) for n = 9..451</a>
%F A320983 a(n) = A262134(n) - A262133(n).
%p A320983 b:= proc(u, o, c) option remember; `if`(c<0 or c>9, 0, `if`(u+o=0,
%p A320983        x^c, (p-> add(coeff(p, x, i)*x^max(i, c), i=0..9))(add(
%p A320983        b(u-j, o-1+j, c-1), j=1..u)+add(b(u+j-1, o-j, c+1), j=1..o))))
%p A320983     end:
%p A320983 a:= n-> coeff(add(b(j-1, n-j, 0), j=1..n), x, 9):
%p A320983 seq(a(n), n=9..30);
%Y A320983 Column k=9 of A262125.
%Y A320983 Cf. A262133, A262134.
%K A320983 nonn
%O A320983 9,3
%A A320983 _Alois P. Heinz_, Oct 25 2018