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A337165
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Number T(n,k) of compositions of n into k nonzero squares; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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25
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1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 3, 0, 0, 1, 0, 0, 0, 0, 4, 0, 0, 1, 0, 0, 1, 0, 0, 5, 0, 0, 1, 0, 1, 0, 3, 0, 0, 6, 0, 0, 1, 0, 0, 2, 0, 6, 0, 0, 7, 0, 0, 1, 0, 0, 0, 3, 0, 10, 0, 0, 8, 0, 0, 1, 0, 0, 0, 1, 4, 0, 15, 0, 0, 9, 0, 0, 1
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OFFSET
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0,18
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LINKS
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FORMULA
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G.f. of column k: (Sum_{j>=1} x^(j^2))^k.
Sum_{k=0..n} k * T(n,k) = A281704(n).
Sum_{k=0..n} (-1)^k * T(n,k) = A317665(n).
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EXAMPLE
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Triangle T(n,k) begins:
1;
0, 1;
0, 0, 1;
0, 0, 0, 1;
0, 1, 0, 0, 1;
0, 0, 2, 0, 0, 1;
0, 0, 0, 3, 0, 0, 1;
0, 0, 0, 0, 4, 0, 0, 1;
0, 0, 1, 0, 0, 5, 0, 0, 1;
0, 1, 0, 3, 0, 0, 6, 0, 0, 1;
0, 0, 2, 0, 6, 0, 0, 7, 0, 0, 1;
0, 0, 0, 3, 0, 10, 0, 0, 8, 0, 0, 1;
0, 0, 0, 1, 4, 0, 15, 0, 0, 9, 0, 0, 1;
...
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MAPLE
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b:= proc(n) option remember; `if`(n=0, 1, add((s->
`if`(s>n, 0, expand(x*b(n-s))))(j^2), j=1..isqrt(n)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..14);
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MATHEMATICA
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b[n_] := b[n] = If[n == 0, 1, Sum[With[{s = j^2},
If[s>n, 0, Expand[x*b[n - s]]]], {j, 1, Sqrt[n]}]];
T[n_] := CoefficientList[b[n], x];
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CROSSREFS
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Columns k=0-10 give: A000007, A010052, A063725, A063691, A063730, A340481, A340905, A340906, A340915, A340946, A340947.
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KEYWORD
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AUTHOR
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STATUS
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approved
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