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A103198
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Number of compositions of n into a square number of parts.
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10
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1, 1, 1, 1, 2, 5, 11, 21, 36, 58, 94, 166, 331, 716, 1574, 3368, 6892, 13447, 25127, 45391, 80428, 142615, 259085, 491855, 982400, 2045001, 4352661, 9291361, 19609786, 40574017, 81973315, 161568281, 311062991, 586764281, 1089615033, 2005257849, 3688711427
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OFFSET
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0,5
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COMMENTS
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Also the number of ways to fill a square matrix with the parts of an integer partition of n. For example, the a(6) = 11 matrices are:
[6]
.
[1 1] [1 1] [1 3] [3 1] [1 1] [1 2] [1 2] [2 1] [2 1] [2 2]
[1 3] [3 1] [1 1] [1 1] [2 2] [1 2] [2 1] [1 2] [2 1] [1 1]
(End)
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LINKS
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FORMULA
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a(n) = Sum_{k>=0} (x/(1-x))^(k^2).
Binomial transform of the characteristic function of squares A010052, with 0th term omitted. - Carl Najafi, Sep 09 2011
a(n) = Sum_{k >= 0} binomial(n-1,k^2-1). - Gus Wiseman, Jan 17 2019
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MAPLE
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b:= proc(n, t) option remember; `if`(n=0,
`if`(issqr(t), 1, 0), add(b(n-j, t+1), j=1..n))
end:
a:= n-> b(n, 0):
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MATHEMATICA
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nmax = 40; Rest[CoefficientList[Series[-1/2 + EllipticTheta[3, 0, x/(1-x)]/2, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jan 03 2017 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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