A008968 - OEIS

A008968

Triangle of distribution of rank sums: Wilcoxon's statistic.

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, 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, 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

(list; graph; refs; listen; history; text; internal format)

OFFSET

6,3

REFERENCES

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 237.

LINKS

Table of n, a(n) for n=6..99.

FORMULA

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).

EXAMPLE

Rows begin:

{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, 4, 3, 2, 1, 1},

{1, 1, 2, 3, 4, 5, 7, 7, 8, 8, 8, 7, 7, 5, 4, 3, 2, 1, 1},

...

MATHEMATICA

f[r_] := Product[(x^i - x^(r+1))/(1 - x^i), {i, 1, r-3}]/x^((r-2)*(r-3)/2);

row[r_] := CoefficientList[ Series[f[r], {x, 0, 3r+1}], x];

Table[row[r], {r, 6, 12}] // Flatten (* Jean-François Alcover, Nov 30 2012 *)

CROSSREFS