login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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,look,tabf

AUTHOR

Thomas Dybdahl Ahle, Apr 07 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 2 13:28 EST 2016. Contains 278678 sequences.