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A320591
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Expansion of Product_{k>=1} (1 + x^k/(1 + x)^k).
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4
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1, 1, 0, 1, -2, 4, -7, 11, -16, 23, -36, 65, -129, 256, -473, 772, -1028, 835, 776, -5755, 17562, -41750, 86678, -165145, 299949, -541837, 1020029, -2068203, 4509512, -10252952, 23465297, -52762788, 115160832, -243018459, 496094524, -982431070, 1894710043, -3574095362
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OFFSET
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0,5
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LINKS
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FORMULA
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G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*((1 + x)^k - x^k))).
G.f.: exp(Sum_{k>=1} A000593(k)*x^k/(k*(1 + x)^k)).
From _Peter Bala_, Dec 22 2020: (Start)
O.g.f.: Sum_{n >= 0} x^(n*(n+1)/2)/Product_{k = 1..n} ((1 + x)^k - x^k). Cf. A307548.
Conjectural o.g.f.: (1/2) * Sum_{n >= 0} x^(n*(n-1)/2)*(1 + x)^n/( Product_{k = 1..n} ( (1 + x)^k - x^k ) ). (End)
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MAPLE
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seq(coeff(series(mul((1+x^k/(1+x)^k), k=1..n), x, n+1), x, n), n = 0 .. 37); # _Muniru A Asiru_, Oct 16 2018
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MATHEMATICA
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nmax = 37; CoefficientList[Series[Product[(1 + x^k/(1 + x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 37; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k/d + 1) d, {d, Divisors[k]}] x^k/(k (1 + x)^k), {k, 1, nmax}]], {x, 0, nmax}], x]
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PROG
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(PARI) m=50; x='x+O('x^m); Vec(prod(k=1, m+2, (1 + x^k/(1 + x)^k))) \\ _G. C. Greubel_, Oct 29 2018
(Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1 + x^k/(1 + x)^k): k in [1..(m+2)]]) )); // _G. C. Greubel_, Oct 29 2018
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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_Ilya Gutkovskiy_, Oct 16 2018
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STATUS
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approved
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