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A320976 Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of two.

%I A320976 #6 Oct 25 2018 16:05:23
%S A320976 0,1,3,24,101,862,4743,47216,322039,3744368,30517747,409498400,
%T A320976 3884904379,59360223088,642766195887,11046815693568,134468538125519,
%U A320976 2571506053105408,34764547687430955,732881798335913984,10895866774781276947,251184536044504689152
%N A320976 Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of two.
%H A320976 Alois P. Heinz, <a href="/A320976/b320976.txt">Table of n, a(n) for n = 2..464</a>
%F A320976 a(n) = A262126(n) - A000111(n).
%p A320976 b:= proc(u, o, c) option remember; `if`(c<0 or c>2, 0, `if`(u+o=0,
%p A320976        x^c, (p-> add(coeff(p, x, i)*x^max(i, c), i=0..2))(add(
%p A320976        b(u-j, o-1+j, c-1), j=1..u)+add(b(u+j-1, o-j, c+1), j=1..o))))
%p A320976     end:
%p A320976 a:= n-> coeff(add(b(j-1, n-j, 0), j=1..n), x, 2):
%p A320976 seq(a(n), n=2..30);
%Y A320976 Column k=2 of A262125.
%Y A320976 Cf. A000111, A262126.
%K A320976 nonn
%O A320976 2,3
%A A320976 _Alois P. Heinz_, Oct 25 2018