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%I #4 Sep 14 2015 15:48:55
%S 1,1,2,5,20,87,522,3271,26167,214946,2148500,21869553,262040897,
%T 3184440794,44442180413,627992981034,9996086297542,161044694650665,
%U 2877551846402242,52059368659632095,1031291013069584902,20699996793232418643,450130761784158558067
%N Number of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value <= 7.
%H Alois P. Heinz, <a href="/A262169/b262169.txt">Table of n, a(n) for n = 0..451</a>
%F a(n) = A262163(n,7).
%p b:= proc(u, o, c) option remember; `if`(c<0 or c>7, 0, `if`(u+o=0,
%p x^c, (p-> add(coeff(p, x, i)*x^max(i, c), i=0..7))(add(
%p b(u-j, o-1+j, c-1), j=1..u)+add(b(u+j-1, o-j, c+1), j=1..o))))
%p end:
%p a:= n-> (p-> add(coeff(p, x, i), i=0..min(n, 7)))(b(0, n, 0)):
%p seq(a(n), n=0..25);
%Y Column k=7 of A262163.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Sep 13 2015