

A259324


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


0



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
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OFFSET

0,3


LINKS

Table of n, a(n) for n=0..106.
E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301312. See Table I.
E. Fix and J. L. Hodges, Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301312. [Annotated scanned copy]


FORMULA

T(u,m) = T(u,m1)+T(um,m), with initial conditions T(0,m)=1, T(m,1)=u+1.


EXAMPLE

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,
...


MAPLE

A0 := proc(u, m)
option remember;
if u = 0 then
1;
elif u < 0 then
0;
elif m = 1 then
u+1 ;
else
procname(u, m1)+procname(um, m) ;
end if;
end proc:
for d from 1 to 15 do
for m from d to 1 by 1 do
printf("%d, ", A0(dm, 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 (* JeanFrançois Alcover, Apr 05 2020 *)


CROSSREFS

Columns give A002620, A000601, A002621, A002622.
Cf. A137679.
Sequence in context: A195076 A163491 A080772 * A216274 A145111 A104795
Adjacent sequences: A259321 A259322 A259323 * A259325 A259326 A259327


KEYWORD

nonn,tabl


AUTHOR

N. J. A. Sloane, Jun 24 2015


STATUS

approved



