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A061198
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Square table by antidiagonals where T(n,k) is 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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4
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1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 2, 1, 1, 0, 2, 2, 2, 1, 1, 0, 3, 4, 3, 2, 1, 1, 0, 4, 5, 4, 3, 2, 1, 1, 0, 5, 7, 6, 5, 3, 2, 1, 1, 0, 6, 9, 9, 6, 5, 3, 2, 1, 1, 0, 8, 13, 12, 10, 7, 5, 3, 2, 1, 1, 0, 10, 16, 16, 13, 10, 7, 5, 3, 2, 1, 1, 0, 12, 22, 22, 19, 14, 11, 7, 5, 3, 2, 1, 1, 0, 15, 27, 29, 25, 20, 14, 11, 7, 5, 3, 2, 1, 1
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OFFSET
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0,12
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LINKS
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FORMULA
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G.f. for row n of table: Product_{j>=1} Sum_{k=0..n} x^(j*k) = Product_{j>=1} (1-x^((n+1)*j)) / (1-x^j). - Sean A. Irvine, Jan 26 2023
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EXAMPLE
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Square table T(n,k) begins:
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, ...
1, 1, 2, 2, 4, 5, 7, 9, 13, 16, 22, ...
1, 1, 2, 3, 4, 6, 9, 12, 16, 22, 29, ...
1, 1, 2, 3, 5, 6, 10, 13, 19, 25, 34, ...
1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, ...
1, 1, 2, 3, 5, 7, 11, 14, 21, 28, 39, ...
1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, ...
1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, ...
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, ...
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...
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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:
A:= (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]}]]];
A[n_, k_] := b[k, k, n];
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CROSSREFS
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A061199 is the same table but excluding 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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