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A325940
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Expansion of Sum_{k>=1} x^(2*k) / (1 + x^k)^2.
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9
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0, 1, -2, 4, -4, 4, -6, 11, -10, 6, -10, 18, -12, 8, -20, 26, -16, 13, -18, 28, -28, 12, -22, 48, -28, 14, -36, 38, -28, 24, -30, 57, -44, 18, -44, 62, -36, 20, -52, 74, -40, 32, -42, 58, -72, 24, -46, 110, -54, 31, -68, 68, -52, 40, -68, 100, -76, 30, -58, 116
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OFFSET
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1,3
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LINKS
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FORMULA
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G.f.: Sum_{k>=2} (-1)^k * (k - 1) * x^k / (1 - x^k).
a(n) = Sum_{d|n} (-1)^d * (d - 1).
Faster converging series: A(q) = Sum_{n >= 1} (-1)^n*q^(n^2)*((n-1)*q^(3*n) + n*q^(2*n) + (n-2)*q^n + n-1)/((1 + q^n)*(1 - q^(2*n))) - apply the operator t*d/dt to equation 1 in Arndt, then set t = -q and x = q. - Peter Bala, Jan 22 2021
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MATHEMATICA
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nmax = 60; CoefficientList[Series[Sum[x^(2 k)/(1 + x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[(-1)^d (d - 1), {d, Divisors[n]}], {n, 1, 60}]
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PROG
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(Magma)
A325940:= func< n | (&+[0^(n mod j)*(-1)^j*(j-1): j in [1..n]]) >;
(SageMath)
def A325940(n): return sum(0^(n%j)*(-1)^j*(j-1) for j in range(1, n+1))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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