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A188668 Triangle read by rows: row n gives (coefficients * n!) in expansion of pieces k=0..n-1 of the cumulative distribution function for the Irwin-Hall distribution, lowest powers first. 2
0, 1, 0, 0, 1, -2, 4, -1, 0, 0, 0, 1, 3, -9, 9, -2, -21, 27, -9, 1, 0, 0, 0, 0, 1, -4, 16, -24, 16, -3, 92, -176, 120, -32, 3, -232, 256, -96, 16, -1, 0, 0, 0, 0, 0, 1, 5, -25, 50, -50, 25, -4, -315, 775, -750, 350, -75, 6, 2115, -3275, 1950, -550, 75, -4, -3005, 3125, -1250, 250, -25, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

This is the probability distribution for the sum of n independent, random variables, each uniformly distributed on [0,1).

REFERENCES

Hall, Philip. (1927) "The Distribution of Means for Samples of Size N Drawn from a Population in which the Variate Takes Values Between 0 and 1, All Such Values Being Equally Probable". Biometrika, Vol. 19, No. 3/4., pp. 240-245.

LINKS

Alois P. Heinz, Rows n = 1..31, flattened

Wikipedia, Irwin-Hall distribution

FORMULA

G.f. for piece k in row n: (1/n!) * Sum_{j=0..k} (-1)^j * C(n,j) * (x-j)^n.

EXAMPLE

For n = 3, k = 2 (three variables, third piece) the distribution is the polynomial: 1/6 * (1*(x-0)^3 - 3*(x-1)^3 + 3*(x-2)^3) = 1/6 * (-21 + 27*x - 9*x^2 + x^3). That gives the subsequence [-21, 27, -9, 1].

Triangle begins:

[0, 1];

[0, 0, 1], [-2, 4, -1];

[0, 0, 0, 1], [3, -9, 9, -2], [-21, 27, -9, 1];

MAPLE

f:= proc(n, k) option remember;

       add ((-1)^j * binomial(n, j) * (x-j)^n, j=0..k)

    end:

T:= (n, k)-> seq (coeff (f(n, k), x, t), t=0..n):

seq (seq (T(n, k), k=0..n-1), n=1..7);  # Alois P. Heinz, Apr 09 2011

CROSSREFS

Sequence in context: A012710 A009512 A163259 * A055599 A115407 A010586

Adjacent sequences:  A188665 A188666 A188667 * A188669 A188670 A188671

KEYWORD

sign,tabf

AUTHOR

Thomas Dybdahl Ahle, Apr 07 2011

STATUS

approved

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Last modified April 19 05:22 EDT 2014. Contains 240738 sequences.