

A188668


Triangle read by rows: row n gives (coefficients * n!) in expansion of pieces k=0..n1 of the cumulative distribution function for the IrwinHall 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
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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. 240245.


LINKS

Alois P. Heinz, Rows n = 1..31, flattened
Wikipedia, IrwinHall distribution


FORMULA

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


EXAMPLE

For n = 3, k = 2 (three variables, third piece) the distribution is the polynomial: 1/6 * (1*(x0)^3  3*(x1)^3 + 3*(x2)^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) * (xj)^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..n1), 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,look,tabf


AUTHOR

Thomas Dybdahl Ahle, Apr 07 2011


STATUS

approved



