|
%I #16 Dec 16 2021 22:35:39
%S 1,1,2,3,3,3,3,2,1,1,1,1,2,3,4,4,5,4,4,3,2,1,1,1,1,2,3,4,5,6,6,6,6,5,
%T 4,3,2,1,1,1,1,2,3,4,5,7,7,8,8,8,7,7,5,4,3,2,1,1,1,1,2,3,4,5,7,8,9,10,
%U 10,10,10,9,8,7,5,4,3,2,1,1,1,1,2,3,4,5,7,8,10,11,12,12,13,12
%N Triangle of distribution of rank sums: Wilcoxon's statistic.
%D F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 237.
%F Let f(r) = Product( (x^i-x^(r+1))/(1-x^i), i = 1..r-3) / x^((r-2)*(r-3)/2); then expanding f(r) in powers of x and taking coefficients gives the successive rows of this triangle (with a different offset).
%e Rows begin:
%e {1, 1, 2, 3, 3, 3, 3, 2, 1, 1},
%e {1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1},
%e {1, 1, 2, 3, 4, 5, 6, 6, 6, 6, 5, 4, 3, 2, 1, 1},
%e {1, 1, 2, 3, 4, 5, 7, 7, 8, 8, 8, 7, 7, 5, 4, 3, 2, 1, 1},
%e ...
%t f[r_] := Product[(x^i - x^(r+1))/(1 - x^i), {i, 1, r-3}]/x^((r-2)*(r-3)/2);
%t row[r_] := CoefficientList[ Series[f[r], {x, 0, 3r+1}], x];
%t Table[row[r], {r, 6, 12}] // Flatten (* _Jean-François Alcover_, Nov 30 2012 *)
%K tabf,nonn,nice
%O 6,3
%A _N. J. A. Sloane_
|
