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A329265
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a(n) is the number of solutions of the infinite Diophantine equation Sum_{j>0} j^r*(k_j)^2 = n with k_j integers and r = 3.
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2
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1, 2, 0, 0, 2, 0, 0, 0, 2, 6, 0, 0, 4, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 4, 2, 0, 2, 4, 0, 0, 4, 2, 8, 0, 4, 18, 0, 0, 8, 0, 4, 0, 4, 12, 0, 0, 0, 4, 2, 0, 8, 4, 0, 0, 0, 0, 8, 0, 4, 16, 0, 0, 12, 4, 4, 0, 0, 16, 0, 0, 8, 10, 16, 0, 8, 16, 0, 0, 0, 4, 18, 0, 0, 16
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = [q^n] Product_{j>0} Product_{n>0} (1 - (-1)^n*q^(n*j^r)) / (1 + (-1)^n*q^(n*j^r)) with r = 3 (see Proposition 1.1 in Zhou and Sun).
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EXAMPLE
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a(9) = 6 since there are 6 integer solutions to 1^3*k1^2 + 2^3*k2^2 + ... = 9:
k1 = +-3 and k_j = 0 for j > 1;
k1 = -1, k2 = +-1 and k_j = 0 for j > 2;
k1 = 1, k2 = +-1 and k_j = 0 for j > 2.
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MATHEMATICA
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nmax=85; r=3; CoefficientList[Series[Product[Product[(1-(-1)^n*q^(n*j^r))/(1+(-1)^n*q^(n*j^r)), {n, 1, nmax}], {j, 1, nmax}], {q, 0, nmax}], q]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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