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%I #24 Jul 19 2024 11:15:31
%S 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,
%T 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,
%U 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
%N Infinite square array read by antidiagonals: T(n,k) = number of ways of partitioning numbers <= n into k parts (n >= 0, k >= 1).
%H E. Fix and J. L. Hodges, Jr., <a href="http://www.jstor.org/stable/2236885">Significance probabilities of the Wilcoxon test</a>, Annals Math. Stat., 26 (1955), 301-312. See Table I.
%H E. Fix and J. L. Hodges, <a href="/A000601/a000601.pdf">Significance probabilities of the Wilcoxon test</a>, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]
%F T(u,m) = T(u,m-1)+T(u-m,m), with initial conditions T(0,m)=1, T(m,1)=u+1.
%e The first few antidiagonals are:
%e 1,
%e 1,2,
%e 1,2,3,
%e 1,2,4,4,
%e 1,2,4,6,5,
%e 1,2,4,7,9,6,
%e 1,2,4,7,11,12,7,
%e 1,2,4,7,12,16,16,8,
%e ...
%p A259324 := proc(u,m)
%p option remember;
%p if u = 0 then
%p 1;
%p elif u < 0 then
%p 0;
%p elif m = 1 then
%p u+1 ;
%p else
%p procname(u,m-1)+procname(u-m,m) ;
%p end if;
%p end proc:
%p for d from 1 to 15 do
%p for m from d to 1 by -1 do
%p printf("%d,",A259324(d-m,m)) ;
%p end do:
%p end do: # _R. J. Mathar_, Jul 14 2015
%t 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;
%t Table[T[u - m, m], {u, 0, 14}, {m, u, 1, -1}] // Flatten (* _Jean-François Alcover_, Apr 05 2020 *)
%Y Columns give A002620, A000601, A002621, A002622.
%Y Cf. A137679.
%K nonn,tabl
%O 0,3
%A _N. J. A. Sloane_, Jun 24 2015
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