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A329264
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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 = 2.
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2
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1, 2, 0, 0, 4, 4, 0, 0, 4, 4, 4, 0, 0, 12, 8, 0, 6, 16, 4, 0, 16, 8, 8, 0, 8, 24, 20, 0, 0, 52, 24, 0, 12, 32, 28, 8, 24, 12, 48, 16, 24, 68, 48, 8, 16, 96, 32, 16, 8, 68, 96, 32, 40, 68, 128, 32, 80, 88, 76, 48, 32, 156, 104, 64, 8, 224, 192, 40, 88, 152, 208
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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 = 2 (see Proposition 1.1 in Zhou and Sun).
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EXAMPLE
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a(16) = 6 since there are 6 integer solutions to 1^2*k1^2 + 2^2*k2^2 + 3^2*k3^2 + 4^2*k4^2 + ... = 16:
k1 = +-4 and k_j = 0 for j > 1;
k1 = 0, k2 = +-2 and k_j = 0 for j > 2;
k1 = k2 = k3 = 0, k4 = +-1 and k_j = 0 for j > 4.
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MATHEMATICA
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nmax=70; r=2; 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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