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.

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

Offset

1,6

Comments

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

Links

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

Philip Hall, 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. (Dec., 1927), pp. 240-245.

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

Mathematica

f[n_, k_] := f[n, k] = Sum[(-1)^j*Binomial[n, j]*(x-j)^n, {j, 0, k}]; T[n_, k_] := Table[Coefficient[f[n, k], x, t], {t, 0, n}]; Table[T[n, k], {n, 1, 7}, { k, 0, n-1}] // Flatten (* Jean-François Alcover, Feb 26 2017, after Alois P. Heinz *)

Crossrefs

Sequence in context: A365951 A163259 A309785 * A366463 A366466 A365948

Adjacent sequences: A188665 A188666 A188667 * A188669 A188670 A188671

Keyword

sign,look,tabf

Author

Thomas Dybdahl Ahle, Apr 07 2011

Status

approved