A050392
Exponential reversion of Euler totient function A000010.
Original entry on oeis.org
1, -1, 1, 3, -39, 257, -909, -6389, 183715, -2326009, 15050003, 140089725, -6804608381, 130909360315, -1286161585477, -12952744700713, 970148927462835, -25588194678272039, 347909302401071797
Offset: 1
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length = 20; Range[length]! InverseSeries[Sum[EulerPhi[n] x^n/n!, {n, 1, length}] + O[x]^(length+1)][[3]] (* Vladimir Reshetnikov, Nov 07 2015 *)
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seq(n)= Vec(serlaplace(serreverse(sum(k=1, n, eulerphi(k)*x^k/k!) + O(x*x^n)))); \\ Michel Marcus, Apr 21 2020
A292875
Expansion of the series reversion of Sum_{k>=1} x^k*k*Product_{p|k, p prime} (1 + 1/p).
Original entry on oeis.org
1, -3, 14, -81, 528, -3708, 27388, -209739, 1650204, -13258230, 108311352, -896946048, 7512187398, -63520243398, 541511083648, -4649182740159, 40163784583752, -348870785898510, 3045109181792304, -26694854975488554, 234936349043049246, -2074958037081265050
Offset: 1
A307487
G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} mu(k)*x^k*A(x)^k/(1 - x^k*A(x)^k)^2, where mu() is the Möbius function (A008683).
Original entry on oeis.org
1, 1, 2, 6, 19, 65, 231, 847, 3187, 12223, 47610, 187836, 749055, 3014453, 12226718, 49931342, 205133243, 847224291, 3515681010, 14650664552, 61286007817, 257256430363, 1083272333869, 4574656128903, 19369837160689, 82214738381631, 349743277470990
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 65*x^5 + 231*x^6 + 847*x^7 + 3187*x^8 + 12223*x^9 + 47610*x^10 + ...
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terms = 27; CoefficientList[1/x InverseSeries[Series[x/(1 + Sum[EulerPhi[k] x^k, {k, 1, terms}]), {x, 0, terms}], x], x]
terms = 27; A[] = 0; Do[A[x] = 1 + Sum[MoebiusMu[k] x^k A[x]^k/(1 - x^k A[x]^k)^2, {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
terms = 27; A[] = 0; Do[A[x] = 1 + Sum[EulerPhi[k] x^k A[x]^k, {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
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