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A188816
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Triangle read by rows: row n gives (coefficients * (n-1)!) in expansion of pieces k=0..n-1 of the probability mass function for the Irwin-Hall distribution, lowest powers first.
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1
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1, 0, 1, 2, -1, 0, 0, 1, -3, 6, -2, 9, -6, 1, 0, 0, 0, 1, 4, -12, 12, -3, -44, 60, -24, 3, 64, -48, 12, -1, 0, 0, 0, 0, 1, -5, 20, -30, 20, -4, 155, -300, 210, -60, 6, -655, 780, -330, 60, -4, 625
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OFFSET
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1,4
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COMMENTS
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This is the probability distribution for the sum of n independent, random variables, each uniformly distributed on [0,1).
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LINKS
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FORMULA
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G.f. for piece k in row n: (1/(n-1)!) * Sum_{j=0..k} (-1)^j * C(n,j) * (x-j)^(n-1).
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EXAMPLE
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For n = 4, k = 1 (four variables, second piece) the function is the polynomial: 1/6 * (4 - 12x + 12x^2 -3x^3). That gives the subsequence [4, -12, 12, -3].
Triangle begins:
[1];
[0,1], [2,-1];
[0,0,1], [-3,6,-2], [9,-6,1];
...
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MAPLE
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f:= proc(n, k) option remember;
add((-1)^j * binomial(n, j) * (x-j)^(n-1), j=0..k)
end:
T:= (n, k)-> seq(coeff(f(n, k), x, t), t=0..n-1):
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MATHEMATICA
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f[n_, k_] := f[n, k] = Sum[(-1)^j Binomial[n, j] (x-j)^(n-1), {j, 0, k}];
T[n_, k_] := Table[Coefficient[f[n, k], x, t], {t, 0, n-1}];
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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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