A259324 Infinite square array read by antidiagonals: T(n,k) = number of ways of partitioning numbers <= n into k parts (n >= 0, k >= 1).

1, 1, 2, 1, 2, 3, 1, 2, 4, 4, 1, 2, 4, 6, 5, 1, 2, 4, 7, 9, 6, 1, 2, 4, 7, 11, 12, 7, 1, 2, 4, 7, 12, 16, 16, 8, 1, 2, 4, 7, 12, 18, 23, 20, 9, 1, 2, 4, 7, 12, 19, 27, 31, 25, 10, 1, 2, 4, 7, 12, 19, 29, 38, 41, 30, 11, 1, 2, 4, 7, 12, 19, 30, 42, 53, 53, 36, 12, 1, 2, 4, 7, 12, 19, 30, 44, 60, 71, 67, 42, 13, 1, 2, 4, 7, 12, 19, 30, 45, 64, 83, 94, 83, 49, 14, 1, 2

Offset: 0

Author

N. J. A. Sloane, Jun 24 2015

Keywords

• nonn
• tabl

Examples

			The first few antidiagonals are:
1,
1,2,
1,2,3,
1,2,4,4,
1,2,4,6,5,
1,2,4,7,9,6,
1,2,4,7,11,12,7,
1,2,4,7,12,16,16,8,
...
		

Links

• E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312. See Table I.
• E. Fix and J. L. Hodges, Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]

Crossrefs

Columns give A002620, A000601, A002621, A002622.

Cf. A137679.

Programs

Maple

A259324 := proc(u,m)
    option remember;
    if u = 0 then
        1;
    elif u < 0 then
        0;
    elif m = 1 then
        u+1 ;
    else
        procname(u,m-1)+procname(u-m,m) ;
    end if;
end proc:
for d from 1 to 15 do
    for m from d to 1 by -1 do
        printf("%d,",A259324(d-m,m)) ;
    end do:
end do: # R. J. Mathar, Jul 14 2015

Mathematica

T[0, ] = 1; T[u /; u > 0, m_ /; m > 1] := T[u, m] = T[u, m - 1] + T[u - m, m]; T[u_, 1] := u + 1; T[, ] = 0;
Table[T[u - m, m], {u, 0, 14}, {m, u, 1, -1}] // Flatten (* Jean-François Alcover, Apr 05 2020 *)

Formula

T(u,m) = T(u,m-1)+T(u-m,m), with initial conditions T(0,m)=1, T(m,1)=u+1.