OFFSET
1,4
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..32, 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-1)!) * Sum_{j=0..k} (-1)^j * C(n,j) * (x-j)^(n-1).
EXAMPLE
For n = 4, k = 1 (four variables, second piece) the function is the polynomial: 1/6 * (4 - 12x + 12x^2 -3x^3). That gives the subsequence [4, -12, 12, -3].
Triangle begins:
[1];
[0,1], [2,-1];
[0,0,1], [-3,6,-2], [9,-6,1];
...
MAPLE
f:= proc(n, k) option remember;
add((-1)^j * binomial(n, j) * (x-j)^(n-1), j=0..k)
end:
T:= (n, k)-> seq(coeff(f(n, k), x, t), t=0..n-1):
seq(seq(T(n, k), k=0..n-1), n=1..7); # Alois P. Heinz, Jul 06 2017
MATHEMATICA
f[n_, k_] := f[n, k] = Sum[(-1)^j Binomial[n, j] (x-j)^(n-1), {j, 0, k}];
T[n_, k_] := Table[Coefficient[f[n, k], x, t], {t, 0, n-1}];
Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 7}] // Flatten (* Jean-François Alcover, Feb 19 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
AUTHOR
Thomas Dybdahl Ahle, Apr 11 2011
STATUS
approved