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A320984 Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of ten.

%I A320984 #4 Oct 25 2018 19:36:14
%S A320984 0,1,11,8220,98827,14052474,185726938,15068990276,219771532102,
%T A320984 13747965316306,220995092995233,12094333663749818,213504732754725133,
%U A320984 10896908531544406283,210288454755592374452,10376111829436767498058,217839166931637914375624
%N A320984 Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of ten.
%H A320984 Alois P. Heinz, <a href="/A320984/b320984.txt">Table of n, a(n) for n = 10..451</a>
%F A320984 a(n) = A262135(n) - A262134(n).
%p A320984 b:= proc(u, o, c) option remember; `if`(c<0 or c>10, 0, `if`(u+o=0,
%p A320984        x^c, (p-> add(coeff(p, x, i)*x^max(i, c), i=0..10))(add(
%p A320984        b(u-j, o-1+j, c-1), j=1..u)+add(b(u+j-1, o-j, c+1), j=1..o))))
%p A320984     end:
%p A320984 a:= n-> coeff(add(b(j-1, n-j, 0), j=1..n), x, 10):
%p A320984 seq(a(n), n=10..30);
%Y A320984 Column k=10 of A262125.
%Y A320984 Cf. A262134, A262135.
%K A320984 nonn
%O A320984 10,3
%A A320984 _Alois P. Heinz_, Oct 25 2018