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A281541
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Expansion of Sum_{i>=1} x^(i^2)/(1 - x^(i^2)) / Product_{j>=1} (1 - x^(j^2)).
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8
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1, 2, 3, 5, 7, 9, 11, 15, 19, 23, 27, 34, 41, 47, 53, 64, 75, 86, 96, 113, 129, 145, 159, 182, 206, 229, 252, 284, 318, 349, 380, 423, 468, 513, 555, 616, 676, 736, 791, 869, 949, 1026, 1103, 1202, 1310, 1408, 1506, 1631, 1766, 1896, 2020, 2185, 2354, 2525, 2680, 2882, 3094, 3305, 3506, 3751, 4023, 4281, 4537
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OFFSET
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1,2
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COMMENTS
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Total number of parts in all partitions of n into squares.
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LINKS
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FORMULA
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G.f.: Sum_{i>=1} x^(i^2)/(1 - x^(i^2)) / Product_{j>=1} (1 - x^(j^2)).
a(n) ~ exp(3 * 2^(-4/3) * zeta(3/2)^(2/3) * (Pi*n)^(1/3)) * sqrt(Pi/3) / (12*sqrt(n)). - Vaclav Kotesovec, Sep 15 2021
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EXAMPLE
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a(8) = 15 because we have [4, 4], [4, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1] and 2 + 5 + 8 = 15.
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n], (s->
`if`(s>n, 0, (p->p+[0, p[1]])(b(n-s, i))))(i^2)+b(n, i-1))
end:
a:= n-> b(n, isqrt(n))[2]:
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MATHEMATICA
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nmax = 63; Rest[CoefficientList[Series[Sum[x^i^2/(1 - x^i^2), {i, 1, nmax}]/Product[1 - x^j^2, {j, 1, nmax}], {x, 0, nmax}], x]]
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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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