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%I #30 Feb 19 2021 07:23:56
%S 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,
%T 16,-3,92,-176,120,-32,3,-232,256,-96,16,-1,0,0,0,0,0,1,5,-25,50,-50,
%U 25,-4,-315,775,-750,350,-75,6,2115,-3275,1950,-550,75,-4,-3005,3125,-1250,250,-25,1
%N 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.
%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="/A188668/b188668.txt">Rows n = 1..31, 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!) * Sum_{j=0..k} (-1)^j * C(n,j) * (x-j)^n.
%e 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].
%e Triangle begins:
%e [0, 1];
%e [0, 0, 1], [-2, 4, -1];
%e [0, 0, 0, 1], [3, -9, 9, -2], [-21, 27, -9, 1];
%e ...
%p f:= proc(n, k) option remember;
%p add((-1)^j * binomial(n, j) * (x-j)^n, j=0..k)
%p end:
%p T:= (n, k)-> seq(coeff(f(n, k), x, t), t=0..n):
%p seq(seq(T(n, k), k=0..n-1), n=1..7); # _Alois P. Heinz_, Apr 09 2011
%t 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_ *)
%K sign,look,tabf
%O 1,6
%A _Thomas Dybdahl Ahle_, Apr 07 2011
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