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A341040
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Number T(n,k) of partitions of n into k distinct nonzero squares; triangle T(n,k), n>=0, 0<=k<=A248509(n), read by rows.
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20
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1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1
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OFFSET
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0
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COMMENTS
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T(n,k) is defined for n, k >= 0. The triangle contains only the terms with 0 <= k <= A248509(n). T(n,k) = 0 for k > A248509(n).
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LINKS
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FORMULA
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T(n,k) = [x^n*y^k] Product_{j>=1} (1 + y*x^(j^2)).
Sum_{k>=0} 2^k * T(n,k) = A279360(n).
Sum_{k>=0} k * T(n,k) = A281542(n).
Sum_{k>=0} (-1)^k * T(n,k) = A276516(n).
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EXAMPLE
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T(62,3) = 2 is the first term > 1 and counts partitions [49,9,4] and [36,25,1].
Triangle T(n,k) begins:
1;
0, 1;
0;
0;
0, 1;
0, 0, 1;
0;
0;
0;
0, 1;
0, 0, 1;
0;
0;
0, 0, 1;
0, 0, 0, 1;
...
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1)+`if`(i^2>n, 0, expand(b(n-i^2, i-1)*x))))
end:
T:= n->(p->seq(coeff(p, x, i), i=0..max(0, degree(p))))(b(n, isqrt(n))):
seq(T(n), n=0..45);
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
b[n, i - 1] + If[i^2 > n, 0, Expand[b[n - i^2, i - 1]*x]]]];
T[n_] := CoefficientList[b[n, Floor@Sqrt[n]], x] /. {} -> {0};
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CROSSREFS
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Columns k=0-10 give: A000007, A010052 (for n>0), A025441, A025442, A025443, A025444, A340988, A340998, A340999, A341000, A341001.
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KEYWORD
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AUTHOR
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STATUS
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approved
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