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A229707
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Triangular array read by rows. T(n,k) is the number of strictly unimodal compositions of n with the greatest part equal to k; n>=1, 1<=k<=n.
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2
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1, 0, 1, 0, 2, 1, 0, 1, 2, 1, 0, 0, 3, 2, 1, 0, 0, 4, 3, 2, 1, 0, 0, 3, 6, 3, 2, 1, 0, 0, 2, 7, 6, 3, 2, 1, 0, 0, 1, 8, 9, 6, 3, 2, 1, 0, 0, 0, 10, 12, 9, 6, 3, 2, 1, 0, 0, 0, 8, 16, 14, 9, 6, 3, 2, 1, 0, 0, 0, 7, 20, 20, 14, 9, 6, 3, 2, 1
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OFFSET
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1,5
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COMMENTS
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A strictly unimodal composition is a composition such that for some j,m 1 <= x(1) < x(2) < ... < x(j) > x(j+1) > ... > x(m) >= 1.
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LINKS
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FORMULA
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O.g.f. for column k: x^k * prod(i=1..k-1, 1 + x^i)^2.
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EXAMPLE
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1,
0, 1,
0, 2, 1,
0, 1, 2, 1,
0, 0, 3, 2, 1,
0, 0, 4, 3, 2, 1,
0, 0, 3, 6, 3, 2, 1,
0, 0, 2, 7, 6, 3, 2, 1,
0, 0, 1, 8, 9, 6, 3, 2, 1,
0, 0, 0, 10, 12, 9, 6, 3, 2, 1
T(7,3) = 3 because we have: 1+2+3+1 = 1+3+2+1 = 2+3+2.
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MAPLE
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b:= proc(n, t, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
`if`(k>0, `if`(n<k, 0, add(b(n-j, j, `if`(j=k, 0, k)),
j=t+1..min(k, n))), add(b(n-j, j, 0), j=1..min(t-1, n))))
end:
T:= (n, k)-> b(n, 0, k):
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MATHEMATICA
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nn=10; Table[Take[Drop[Transpose[Map[PadRight[#, nn+1, 0]&, Table[CoefficientList[Series[x^n Product[(1+x^i), {i, 1, n-1}]^2, {x, 0, nn}], x], {n, 1, nn}]]], 1][[n]], n], {n, 1, nn}]//Grid
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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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