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A262163 Number A(n,k) of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A262163 #13 Oct 26 2018 18:45:32
%S A262163 1,1,0,1,1,0,1,1,1,0,1,1,2,2,0,1,1,2,4,5,0,1,1,2,5,16,16,0,1,1,2,5,19,
%T A262163 54,61,0,1,1,2,5,20,82,324,272,0,1,1,2,5,20,86,454,1532,1385,0,1,1,2,
%U A262163 5,20,87,516,2795,12256,7936,0,1,1,2,5,20,87,521,3135,20346,74512,50521,0
%N A262163 Number A(n,k) of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A262163 Alois P. Heinz, <a href="/A262163/b262163.txt">Antidiagonals n = 0..140, flattened</a>
%F A262163 A(n,k) = Sum_{i=0..k} A258829(n,i).
%e A262163 Square array A(n,k) begins:
%e A262163   1,   1,    1,    1,    1,    1,    1,    1, ...
%e A262163   0,   1,    1,    1,    1,    1,    1,    1, ...
%e A262163   0,   1,    2,    2,    2,    2,    2,    2, ...
%e A262163   0,   2,    4,    5,    5,    5,    5,    5, ...
%e A262163   0,   5,   16,   19,   20,   20,   20,   20, ...
%e A262163   0,  16,   54,   82,   86,   87,   87,   87, ...
%e A262163   0,  61,  324,  454,  516,  521,  522,  522, ...
%e A262163   0, 272, 1532, 2795, 3135, 3264, 3270, 3271, ...
%p A262163 b:= proc(u, o, c) option remember; `if`(c<0, 0, `if`(u+o=0, x^c,
%p A262163       (p-> add(coeff(p, x, i)*x^max(i, c), i=0..degree(p)))(add(
%p A262163        b(u-j, o-1+j, c-1), j=1..u)+add(b(u+j-1, o-j, c+1), j=1..o))))
%p A262163     end:
%p A262163 A:= (n, k)-> (p-> add(coeff(p, x, i), i=0..min(n, k)))(b(0, n, 0)):
%p A262163 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A262163 b[u_, o_, c_] := b[u, o, c] = If[c < 0, 0, If[u + o == 0, x^c, Function[p, Sum[Coefficient[p, x, i]*x^Max[i, c], {i, 0, Exponent[p, x]}]][Sum[b[u - j, o - 1 + j, c - 1], {j, 1, u}] + Sum[b[u + j - 1, o - j, c + 1], {j, 1, o}]]]]; A[n_, k_] := Function[p, Sum[Coefficient[p, x, i], {i, 0, Min[n, k]}]][b[0, n, 0]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Feb 22 2016, after _Alois P. Heinz_ *)
%Y A262163 Columns k=0-10 give: A000007, A000111 for n>0, A262164, A262165, A262166, A262167, A262168, A262169, A262170, A262171, A262172.
%Y A262163 Main diagonal gives: A258830.
%Y A262163 Cf. A258829, A262124.
%K A262163 nonn,tabl
%O A262163 0,13
%A A262163 _Alois P. Heinz_, Sep 13 2015