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 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
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS This is the probability distribution for the sum of n independent, random variables, each uniformly distributed on [0,1). REFERENCES Hall, Philip. (1927) "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., pp. 240-245. LINKS Alois P. Heinz, Rows n = 1..32, flattened 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 CROSSREFS Differentiation of A188668. Sequence in context: A127840 A145153 A255517 * A168312 A076837 A055363 Adjacent sequences:  A188813 A188814 A188815 * A188817 A188818 A188819 KEYWORD sign,look,tabf AUTHOR Thomas Dybdahl Ahle, Apr 11 2011 STATUS approved

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Last modified October 15 09:22 EDT 2019. Contains 328026 sequences. (Running on oeis4.)