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

Offset: 6

Author

N. J. A. Sloane

Keywords

• tabf
• nonn
• nice

Examples

			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},
...
		

References

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

Programs

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 *)

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).