A291489
Expansion of the series reversion of -1 + Product_{k>=1} (1 + x^k)^k.
Original entry on oeis.org
1, -2, 3, 2, -41, 196, -541, 229, 7235, -48228, 175956, -254933, -1575661, 14909191, -67194669, 153944915, 292516673, -4968647665, 27275432639, -82747735226, 3883854725, 1660136515050, -11302429310683, 42362000190568, -53376259124482, -520085199830413, 4671353423344131
Offset: 1
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nmax = 27; Rest[CoefficientList[InverseSeries[Series[-1 + Product[(1 + x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x], x]]
nmax = 27; Rest[CoefficientList[InverseSeries[Series[-1 + Exp[Sum[(-1)^(k + 1) x^k/(k (1 - x^k)^2), {k, 1, nmax}]], {x, 0, nmax}], x], x]]
A176950
G.f.: A(x) = 1 + x/Series_Reversion(eta(x) - 1).
Original entry on oeis.org
1, 1, 2, 6, 19, 64, 223, 799, 2927, 10922, 41382, 158800, 615939, 2410880, 9510650, 37774357, 150929671, 606239784, 2446566976, 9915210221, 40336587662, 164662328192, 674300310836, 2769234827610, 11402791485018, 47067085053193
Offset: 1
G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 19*x^5 + 64*x^6 +...
eta(x)-1 = -x - x^2 + x^5 + x^7 - x^12 - x^15 + x^22 + x^26 +...
x/(A(x)-1) = -x - x^2 - 2*x^3 - 5*x^4 - 15*x^5 - 49*x^6 - 169*x^7 -... (cf. A176025).
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Rest[CoefficientList[1 + x/InverseSeries[Series[QPochhammer[x] - 1, {x, 0, 30}]], x]] (* Vaclav Kotesovec, Nov 11 2017 *)
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{a(n)=polcoeff(1+x/serreverse(eta(x+x^2*O(x^n))-1),n)}
A291488
Expansion of the series reversion of -1 + Product_{k>=1} 1/(1 - x^k)^k.
Original entry on oeis.org
1, -3, 12, -58, 318, -1896, 11966, -78595, 531486, -3674324, 25845131, -184348434, 1330147092, -9690872427, 71189146313, -526703176813, 3921274277132, -29354616797397, 220824254874928, -1668453804382315, 12655766723174710, -96340024533522759, 735747052686408916, -5635489764030599334
Offset: 1
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nmax = 24; Rest[CoefficientList[InverseSeries[Series[-1 + Product[1/(1 - x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x], x]]
nmax = 24; Rest[CoefficientList[InverseSeries[Series[-1 + Exp[Sum[DivisorSigma[2, k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x], x]]
A291695
Expansion of the series reversion of Sum_{i>=1} x^i/(1 - x^i) / Product_{j>=1} (1 - x^j).
Original entry on oeis.org
1, -3, 12, -57, 304, -1757, 10746, -68450, 449274, -3016645, 20618317, -142946735, 1002722249, -7103064540, 50738237140, -365049115546, 2642981328372, -19241453032254, 140770867457795, -1034409857616986, 7631075823632553, -56497364856268721, 419641611512419630, -3126180409889288924
Offset: 1
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nmax = 24; Rest[CoefficientList[InverseSeries[Series[Sum[x^i/(1 - x^i), {i, 1, nmax}] / Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x], x]]
nmax = 24; Rest[CoefficientList[InverseSeries[Series[(Log[1-x] + QPolyGamma[0, 1, x]) / (Log[x]*QPochhammer[x]), {x, 0, nmax}], x], x]] (* Vaclav Kotesovec, Apr 21 2020 *)
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