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A259324
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Infinite square array read by antidiagonals: T(n,k) = number of ways of partitioning numbers <= n into k parts (n >= 0, k >= 1).
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0
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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
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0,3
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LINKS
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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), 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]
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FORMULA
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T(u,m) = T(u,m-1)+T(u-m,m), with initial conditions T(0,m)=1, T(m,1)=u+1.
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EXAMPLE
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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,
...
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MAPLE
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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, 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, ", A0(d-m, m)) ;
end do:
end do: # R. J. Mathar, Jul 14 2015
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MATHEMATICA
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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 *)
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CROSSREFS
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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
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KEYWORD
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nonn,tabl
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AUTHOR
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N. J. A. Sloane, Jun 24 2015
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STATUS
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approved
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