A188816 Triangle read by rows: row n gives (coefficients * (n-1)!) in expansion of pieces k=0..n-1 of the probability mass function for the Irwin-Hall distribution, lowest powers first.

1, 0, 1, 2, -1, 0, 0, 1, -3, 6, -2, 9, -6, 1, 0, 0, 0, 1, 4, -12, 12, -3, -44, 60, -24, 3, 64, -48, 12, -1, 0, 0, 0, 0, 1, -5, 20, -30, 20, -4, 155, -300, 210, -60, 6, -655, 780, -330, 60, -4, 625

Offset: 1

Author

Thomas Dybdahl Ahle, Apr 11 2011

Keywords

• sign
• look
• tabf

Comments

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

Examples

			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];
  ...
		

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

Crossrefs

Differentiation of A188668.

Programs

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 *)

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).