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A229892 Number T(n,k) of k up, k down permutations of [n]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A229892 #41 Oct 07 2018 18:20:31
%S A229892 1,1,1,0,1,1,0,2,1,1,0,5,3,1,1,0,16,6,4,1,1,0,61,26,10,5,1,1,0,272,71,
%T A229892 20,15,6,1,1,0,1385,413,125,35,21,7,1,1,0,7936,1456,461,70,56,28,8,1,
%U A229892 1,0,50521,10576,1301,574,126,84,36,9,1,1
%N A229892 Number T(n,k) of k up, k down permutations of [n]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C A229892 T(n,k) is defined for n,k >= 0.  The triangle contains only the terms with k<=n. T(n,k) = T(n,n) = A000012(n) = 1 for k>n.
%C A229892 T(2*n,n) = C(2*n-1,n) = A088218(n) = A001700(n-1) for n>0.
%C A229892 T(2*n+1,n) = C(2*n,n) = A000984(n).
%C A229892 T(2*n+1,n+1) = C(2n,n-1) = A001791(n) for n>0.
%H A229892 Alois P. Heinz, <a href="/A229892/b229892.txt">Rows n = 0..140, flattened</a>
%F A229892 T(7,3) = 20: 1237654, 1247653, 1257643, 1267543, 1347652, 1357642, 1367542, 1457632, 1467532, 1567432, 2347651, 2357641, 2367541, 2457631, 2467531, 2567431, 3457621, 3467521, 3567421, 4567321.
%e A229892 Triangle T(n,k) begins:
%e A229892   1;
%e A229892   1,    1;
%e A229892   0,    1,   1;
%e A229892   0,    2,   1,   1;
%e A229892   0,    5,   3,   1,  1;
%e A229892   0,   16,   6,   4,  1,  1;
%e A229892   0,   61,  26,  10,  5,  1, 1;
%e A229892   0,  272,  71,  20, 15,  6, 1, 1;
%e A229892   0, 1385, 413, 125, 35, 21, 7, 1, 1;
%p A229892 b:= proc(u, o, t, k) option remember; `if`(u+o=0, 1, add(`if`(t=k,
%p A229892        b(o-j, u+j-1, 1, k), b(u+j-1, o-j, t+1, k)), j=1..o))
%p A229892     end:
%p A229892 T:= (n, k)-> `if`(k+1>=n, 1, `if`(k=0, 0, b(0, n, 0, k))):
%p A229892 seq(seq(T(n, k), k=0..n), n=0..10);
%t A229892 b[u_, o_, t_, k_] := b[u, o, t, k] = If[u+o == 0, 1, Sum[If[t == k, b[o-j, u+j-1, 1, k], b[u+j-1, o-j, t+1, k]], {j, 1, o}]]; t[n_, k_] := If[k+1 >= n, 1, If[k == 0, 0, b[0, n, 0, k]]]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Dec 17 2013, translated from Maple *)
%Y A229892 Columns k=1-10 give: A000111, A058258, A229884, A229885, A229886, A229887, A229888, A229889, A229890, A229891.
%Y A229892 Cf. A227941, A229066, A229551.
%K A229892 nonn,tabl,eigen
%O A229892 0,8
%A A229892 _Alois P. Heinz_, Oct 02 2013