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A185282
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Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of n-element subsets that can be chosen from {1,2,...,2*n^k} having element sum n^(k+1).
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3
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1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 3, 0, 1, 1, 7, 36, 7, 0, 1, 1, 15, 351, 785, 18, 0, 1, 1, 31, 3240, 56217, 26404, 51, 0, 1, 1, 63, 29403, 3695545, 18878418, 1235580, 155, 0, 1, 1, 127, 265356, 238085177, 12107973904, 11163952389, 74394425, 486, 0
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OFFSET
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0,13
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COMMENTS
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A(n,k) is the number of partitions of n^(k+1) into n distinct parts <= 2*n^k.
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LINKS
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EXAMPLE
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A(0,0) = 1: {}.
A(1,1) = 1: {1}.
A(2,2) = 3: {1,7}, {2,6}, {3,5}.
A(3,1) = 3: {1,2,6}, {1,3,5}, {2,3,4}.
A(4,1) = 7: {1,2,5,8}, {1,2,6,7}, {1,3,4,8}, {1,3,5,7}, {1,4,5,6}, {2,3,4,7}, {2,3,5,6}.
A(2,3) = 7: {1,15}, {2,14}, {3,13}, {4,12}, {5,11}, {6,10}, {7,9}.
Square array A(n,k) begins:
1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, ...
0, 1, 3, 7, 15, ...
0, 3, 36, 351, 3240, ...
0, 7, 785, 56217, 3695545, ...
0, 18, 26404, 18878418, 12107973904, ...
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MAPLE
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b:= proc(n, i, t) option remember;
`if`(i<t or n<t*(t+1)/2 or n>t*(2*i-t+1)/2, 0,
`if`(n=0, 1, b(n, i-1, t) +`if`(n<i, 0, b(n-i, i-1, t-1))))
end:
A:= (n, k)-> b(n^(k+1), 2*n^k, n):
seq(seq(A(n, d-n), n=0..d), d=0..8);
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MATHEMATICA
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$RecursionLimit = 10000; b[n_, i_, t_] := b[n, i, t] = If [i < t || n < t*(t+1)/2 || n > t*(2*i-t+1)/2, 0, If[n == 0, 1, b[n, i-1, t] + If[n < i, 0, b[n-i, i-1, t-1]]]]; A[0, _] = A[1, _] = 1; A[n_ /; n > 1, 0] = 0; A[n_, k_] := b[n^(k+1), 2*n^k, n]; Table[Print[ta = Table [A[n, d-n], {n, 0, d}]]; ta, {d, 0, 9}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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