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A263789
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Triangle read by rows: T(n,k) (n>=0, 0<=k<=floor(n/2)) is the number of permutations of n and k valleys (considered cyclically).
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7
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1, 1, 0, 2, 0, 6, 0, 16, 8, 0, 40, 80, 0, 96, 528, 96, 0, 224, 2912, 1904, 0, 512, 14592, 23040, 2176, 0, 1152, 69120, 221184, 71424, 0, 2560, 316160, 1858560, 1372160, 79360, 0, 5632, 1413632, 14353152, 20252672, 3891712, 0, 12288, 6223872, 104742912
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OFFSET
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0,4
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COMMENTS
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Conjecture: column k > 0 is asymptotic to n * 2^(n-2*k) * k^(n-1). - Vaclav Kotesovec, Oct 26 2015
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LINKS
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FORMULA
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EXAMPLE
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Triangle begins:
1;
1;
0, 2;
0, 6;
0, 16, 8;
0, 40, 80;
0, 96, 528, 96;
...
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MAPLE
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b:= proc(u, o, t) option remember; expand(`if`(u+o=0, x,
add(b(u-j, o+j-1, 0), j=1..u)*`if`(min(t, n)>0, x, 1)+
add(b(u+j-1, o-j, 1), j=1..o)))
end:
T:= n-> `if`(n<2, 1, (p-> seq(n*coeff(p, x, i)
, i=0..degree(p)))(b(n-1, 0$2))):
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MATHEMATICA
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b[u_, o_, t_] := b[u, o, t] = Expand[If[u+o == 0, x, Sum[b[u-j, o+j-1, 0], {j, 1, u}]*If[Min[t, n] > 0, x, 1] + Sum[b[u+j-1, o-j, 1], {j, 1, o}]]]; T[n_] := If[n<2, 1, Function[p, Table[n*Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n-1, 0, 0]]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 24 2017, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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