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
KEYWORD
AUTHOR
Thomas Dybdahl Ahle, Apr 07 2011
STATUS
approved