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A229892
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Number T(n,k) of k up, k down permutations of [n]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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14
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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, 20, 15, 6, 1, 1, 0, 1385, 413, 125, 35, 21, 7, 1, 1, 0, 7936, 1456, 461, 70, 56, 28, 8, 1, 1, 0, 50521, 10576, 1301, 574, 126, 84, 36, 9, 1, 1
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OFFSET
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0,8
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COMMENTS
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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.
T(2*n+1,n) = C(2*n,n) = A000984(n).
T(2*n+1,n+1) = C(2n,n-1) = A001791(n) for n>0.
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LINKS
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FORMULA
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T(7,3) = 20: 1237654, 1247653, 1257643, 1267543, 1347652, 1357642, 1367542, 1457632, 1467532, 1567432, 2347651, 2357641, 2367541, 2457631, 2467531, 2567431, 3457621, 3467521, 3567421, 4567321.
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EXAMPLE
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Triangle T(n,k) begins:
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, 20, 15, 6, 1, 1;
0, 1385, 413, 125, 35, 21, 7, 1, 1;
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MAPLE
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b:= proc(u, o, t, k) option remember; `if`(u+o=0, 1, add(`if`(t=k,
b(o-j, u+j-1, 1, k), b(u+j-1, o-j, t+1, k)), j=1..o))
end:
T:= (n, k)-> `if`(k+1>=n, 1, `if`(k=0, 0, b(0, n, 0, k))):
seq(seq(T(n, k), k=0..n), n=0..10);
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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