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A061199
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Upper right triangle read by columns where T(n,k), with k >= n, is the number of partitions of k where no part appears more than n times; also partitions of k where no parts are multiples of (n+1).
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7
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1, 0, 1, 0, 1, 2, 0, 2, 2, 3, 0, 2, 4, 4, 5, 0, 3, 5, 6, 6, 7, 0, 4, 7, 9, 10, 10, 11, 0, 5, 9, 12, 13, 14, 14, 15, 0, 6, 13, 16, 19, 20, 21, 21, 22, 0, 8, 16, 22, 25, 27, 28, 29, 29, 30, 0, 10, 22, 29, 34, 37, 39, 40, 41, 41, 42, 0, 12, 27, 38, 44, 49, 51, 53, 54, 55, 55, 56, 0, 15, 36
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OFFSET
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0,6
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LINKS
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EXAMPLE
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T(2,4) = 4 since the possible partitions of 4 are on the first definition (no term more than twice) 1+1+2, 2+2, 1+3, or 4 and on the second definition (no term a multiple of 3) 1+1+1+1, 1+1+2, 2+2, or 4.
Triangle T(n,k) begins:
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 1, 2, 2, 3, 4, 5, 6, 8, ...
2, 2, 4, 5, 7, 9, 13, 16, ...
3, 4, 6, 9, 12, 16, 22, ...
5, 6, 10, 13, 19, 25, ...
7, 10, 14, 20, 27, ...
11, 14, 21, 28, ...
15, 21, 29, ...
22, 29, ...
30, ...
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MAPLE
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b:= proc(n, i, k) option remember;
`if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1, k), j=0..min(n/i, k))))
end:
T:= (n, k)-> b(k$2, n):
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MATHEMATICA
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b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i-1, k], {j, 0, Min[n/i, k]}]]]; T[n_, k_] := b[k, k, n]; Table[Table[T[n, k], {n, 0, k}], {k, 0, 12}] // Flatten (* Jean-François Alcover, Jan 28 2015, after Alois P. Heinz *)
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CROSSREFS
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A061198 is the same table but includes cases where n>k.
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KEYWORD
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AUTHOR
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STATUS
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approved
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