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%I #41 Nov 21 2022 08:57:45
%S 0,0,1,0,1,0,1,1,1,1,2,2,3,3,4,5,7,8,11,13,17,21,28,34,45,56,73,92,
%T 120,151,197,250,324,414,537,687,892,1145,1484,1911,2479,3196,4148,
%U 5359,6954,9000,11687,15140,19672,25516,33166,43065,56010,72784,94716,123185,160380,208740,271913,354123,461529,601436,784209,1022505,1333856
%N Inverse Euler transform of A000931.
%H Joerg Arndt, <a href="/A018243/b018243.txt">Table of n, a(n) for n = 1..1000</a>
%H D. J. Broadhurst and D. Kreimer, <a href="http://arXiv.org/abs/hep-th/9609128">Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops</a>, Phys. Lett. B 393, No.3-4, 403-412 (1997).
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F a(n) = A113788(n) unless n=2. - _Michael Somos_, Apr 06 2012
%F Reciprocal of g.f. of A000931 = (1 - x^2 - x^3) / (1 - x^2) = 1 - x^3 - x^5 - x^7 - x^9 - ... = Product_{k>0} (1 - x^k)^a(n). - _Michael Somos_, Jul 17 2012
%F a(n) ~ A060006^n / n. - _Vaclav Kotesovec_, Oct 09 2019
%e x^3 + x^5 + x^7 + x^8 + x^9 + x^10 + 2*x^11 + 2*x^12 + 3*x^13 + 3*x^14 + ...
%p # The function EulerInvTransform is defined in A358451.
%p a := EulerInvTransform(A000931):
%p seq(a(n), n = 1..65); # _Peter Luschny_, Nov 21 2022
%t a[n_] := (1/n)*Sum[ MoebiusMu[n/d]*Floor[ Re[ N[ RootSum[ -1-#+#^3&, #^d& ]]]] , {d, Divisors[n]}]; a[2]=0; Table[a[n], {n, 1, 65}] (* _Jean-François Alcover_, Oct 05 2012, after _Michael Somos_ *)
%o (Sage)
%o z = PowerSeriesRing(ZZ, 'z').gen().O(30)
%o r = (1 - (z**2 + z**3))/(1 - z**2)
%o F = -z*r.derivative()/r
%o [sum(moebius(n//d)*F[d] for d in divisors(n))//n for n in range(1, 24)] # _F. Chapoton_, Apr 25 2020
%Y Cf. A000931, A113788.
%K nonn,nice
%O 1,11
%A _N. J. A. Sloane_, _David Broadhurst_
%E More terms from _Joerg Arndt_, Jul 18 2012
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