%I #14 Feb 19 2021 07:25:00

%S 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,

%T -48,12,-1,0,0,0,0,1,-5,20,-30,20,-4,155,-300,210,-60,6,-655,780,-330,

%U 60,-4,625

%N 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.

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

%H Alois P. Heinz, <a href="/A188816/b188816.txt">Rows n = 1..32, flattened</a>

%H Philip Hall, <a href="http://www.jstor.org/stable/2331961">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</a>, Biometrika, Vol. 19, No. 3/4. (Dec., 1927), pp. 240-245.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Irwin%E2%80%93Hall_distribution">Irwin-Hall distribution</a>

%F G.f. for piece k in row n: (1/(n-1)!) * Sum_{j=0..k} (-1)^j * C(n,j) * (x-j)^(n-1).

%e 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].

%e Triangle begins:

%e [1];

%e [0,1], [2,-1];

%e [0,0,1], [-3,6,-2], [9,-6,1];

%e ...

%p f:= proc(n, k) option remember;

%p add((-1)^j * binomial(n, j) * (x-j)^(n-1), j=0..k)

%p end:

%p T:= (n, k)-> seq(coeff(f(n, k), x, t), t=0..n-1):

%p seq(seq(T(n, k), k=0..n-1), n=1..7); # _Alois P. Heinz_, Jul 06 2017

%t f[n_, k_] := f[n, k] = Sum[(-1)^j Binomial[n, j] (x-j)^(n-1), {j, 0, k}];

%t T[n_, k_] := Table[Coefficient[f[n, k], x, t], {t, 0, n-1}];

%t Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 7}] // Flatten (* _Jean-François Alcover_, Feb 19 2021, after _Alois P. Heinz_ *)

%Y Differentiation of A188668.

%K sign,look,tabf

%O 1,4

%A _Thomas Dybdahl Ahle_, Apr 11 2011