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A320979 Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of five.

%I A320979 #4 Oct 25 2018 19:18:51
%S A320979 0,1,6,254,1835,47673,416221,9565156,99383961,2250472801,27333591309,
%T A320979 635688426842,8878319017022,215812184750821,3416973303551969,
%U A320979 87455366666951644,1550782738938548075,41903722165381482287,823596208419940694670,23503436481574417378942
%N A320979 Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of five.
%H A320979 Alois P. Heinz, <a href="/A320979/b320979.txt">Table of n, a(n) for n = 5..453</a>
%F A320979 a(n) = A262130(n) - A262129(n).
%p A320979 b:= proc(u, o, c) option remember; `if`(c<0 or c>5, 0, `if`(u+o=0,
%p A320979        x^c, (p-> add(coeff(p, x, i)*x^max(i, c), i=0..5))(add(
%p A320979        b(u-j, o-1+j, c-1), j=1..u)+add(b(u+j-1, o-j, c+1), j=1..o))))
%p A320979     end:
%p A320979 a:= n-> coeff(add(b(j-1, n-j, 0), j=1..n), x, 5):
%p A320979 seq(a(n), n=5..30);
%Y A320979 Column k=5 of A262125.
%Y A320979 Cf. A262129, A262130.
%K A320979 nonn
%O A320979 5,3
%A A320979 _Alois P. Heinz_, Oct 25 2018