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A278340
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Number of partitions of n*(n+1)/2 into distinct squares.
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5
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1, 1, 0, 0, 1, 0, 1, 0, 1, 2, 1, 3, 4, 3, 4, 4, 3, 4, 9, 14, 18, 19, 8, 16, 25, 27, 47, 37, 55, 83, 66, 92, 100, 108, 214, 189, 201, 303, 334, 535, 587, 587, 689, 764, 908, 1278, 1494, 1904, 2369, 2744, 2970, 3269, 3805, 4780, 6701, 7744, 9120, 10582, 11082
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refs;
listen;
history;
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OFFSET
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0,10
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..1000
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FORMULA
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a(n) = [x^(n*(n+1)/2)] Product_{i>=1} (1+x^(i^2)).
a(n) = A033461(A000217(n)).
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EXAMPLE
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a(9) = 2: [25,16,4], [36,9].
a(10) = 1: [25,16,9,4,1].
a(11) = 3: [36,16,9,4,1], [36,25,4,1], [49,16,1].
a(12) = 4: [36,25,16,1], [49,16,9,4], [49,25,4], [64,9,4,1]
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MAPLE
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b:= proc(n, i) option remember; (m-> `if`(n>m, 0,
`if`(n=m, 1, b(n, i-1)+ `if`(i^2>n, 0,
b(n-i^2, i-1)))))(i*(i+1)*(2*i+1)/6)
end:
a:= n-> (m-> b(m, isqrt(m)))(n*(n+1)/2):
seq(a(n), n=0..80);
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MATHEMATICA
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b[n_, i_] := b[n, i] = (If[n > #, 0, If[n == #, 1, b[n, i - 1] + If[i^2 > n, 0, b[n - i^2, i - 1]]]]) &[i*(i + 1)*(2*i + 1)/6];
a[n_] := b[#, Floor @ Sqrt[#]] &[n*(n + 1)/2];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 20 2018, translated from Maple *)
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CROSSREFS
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Cf. A000217, A000290, A033461, A278339.
Sequence in context: A358193 A122530 A301453 * A324749 A022466 A144254
Adjacent sequences: A278337 A278338 A278339 * A278341 A278342 A278343
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KEYWORD
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nonn
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AUTHOR
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Alois P. Heinz, Nov 18 2016
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STATUS
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approved
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