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A163181
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T(n,k) is the number of weak compositions of k into n parts no greater than (n-1) for n>=1, 0<=k<=n(n-1).
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3
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1, 1, 2, 1, 1, 3, 6, 7, 6, 3, 1, 1, 4, 10, 20, 31, 40, 44, 40, 31, 20, 10, 4, 1, 1, 5, 15, 35, 70, 121, 185, 255, 320, 365, 381, 365, 320, 255, 185, 121, 70, 35, 15, 5, 1, 1, 6, 21, 56, 126, 252, 456, 756, 1161, 1666, 2247, 2856, 3431, 3906, 4221, 4332, 4221, 3906, 3431
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OFFSET
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1,3
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COMMENTS
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T(n,k) is the number of length n sequences on an alphabet of {0,1,2,...,n-1} that have a sum of k. Equivalently T(n,k) is the number of functions f:{1,2,...,n}->{0,1,2,...,n-1} such that Sum(f(i)=k, i=1...n).
Row n is also row n of the array of q-nomial coefficients. - Matthew Vandermast, Oct 31 2010
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LINKS
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FORMULA
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O.g.f. for row n is ((1-x^n)/(1-x))^n. For k<=(n-1), T(n,k) = C(n+k-1,k).
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EXAMPLE
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T(3,4) = 6 because there are 6 ternary sequences of length three that sum to 4: [0, 2, 2], [1, 1, 2], [1, 2, 1], [2, 0, 2], [2, 1, 1], [2, 2, 0].
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MAPLE
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b:= proc(n, k, l) option remember; `if`(k=0, 1,
`if`(l=0, 0, add(b(n, k-j, l-1), j=0..min(n-1, k))))
end:
T:= (n, k)-> b(n, k, n):
seq(seq(T(n, k), k=0..n*(n-1)), n=1..8); # Alois P. Heinz, Feb 21 2013
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MATHEMATICA
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(*warning very inefficient*) Table[Distribution[Map[Total, Strings[Range[n], n]]], {n, 1, 6}]//Grid
nn=100; Table[CoefficientList[Series[Sum[x^i, {i, 0, n-1}]^n, {x, 0, nn}], x], {n, 1, 10}]//Grid (* Geoffrey Critzer, Feb 21 2013*)
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CROSSREFS
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The maximum of row n is in column k=n(n-1)/2 = A000217(n-1).
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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