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%I #34 Mar 13 2020 18:06:14
%S 1,-1,0,2,-3,0,5,0,-21,14,117,-342,210,935,-2565,1864,2751,-3945,
%T -8074,4046,108927,-333832,246895,887040,-2764795,3062749,-1372098,
%U 4775900,-9367698,-55130625,299939766,-537241936,-140898285,2464380030,-4060507784,193070394
%N Reversion of g.f. (with constant term included) for partition numbers.
%C See A301624 for the corresponding series reversion for the plane partition numbers A000219. - _Peter Bala_, Feb 09 2020
%H P. Bala, <a href="/A066398/a066398_1.pdf">Representing a sequence as [x^n] G(x)^n</a>
%H <a href="/index/Res#revert">Index entries for reversions of series</a>
%F The o.g.f. A(x) = 1 - x + 2*x^3 - 3*x^4 + 5*x^6 - ... satisfies [x^n](1/A(x))^n = sigma(n) = A000203(n) for n >= 1. - _Peter Bala_, Aug 23 2015
%F G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 - x^k*A(x)^k). - _Ilya Gutkovskiy_, Mar 21 2018
%p with(numtheory):
%p Order := 36:
%p Gser := solve(series(x*exp(add(sigma[1](n)*x^n/n, n = 1..35)), x) = y, x):
%p seq(coeff(Gser, y^k), k = 1..35); # _Peter Bala_, Feb 09 2020
%t nmax = 34; sol = {a[0] -> 1};
%t Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - Product[ 1 - x^k*A[x]^k, {k, 1, n}] + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
%t sol /. Rule -> Set;
%t a /@ Range[0, nmax] (* _Jean-François Alcover_, Nov 02 2019 *)
%Y Cf. A000041, A000203, A007312, A301624.
%K sign
%O 0,4
%A _N. J. A. Sloane_, Dec 25 2001
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