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%I #12 Apr 14 2019 07:51:38
%S 0,1,-2,-2,10,14,-86,-126,858,1302,-9378,-14606,108954,172698,
%T -1319966,-2119118,16489594,26731542,-210887998,-344490170,2747510514,
%U 4515757426,-36336187630,-60023827438,486540793914,807121753178,-6582918170714,-10959656342678,89860260268098
%N G.f. A(x) satisfies: A(x) = x*exp(2*A(-x) + 2*A(-x^3)/3 + 2*A(-x^5)/5 + 2*A(-x^7)/7 + 2*A(-x^9)/9 + ...).
%F G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} ((1 + x^n)/(1 - x^n))^((-1)^n*a(n)).
%F Recurrence: a(n+1) = (2/n) * Sum_{k=1..n} ( Sum_{d|k, k/d odd} (-1)^d*d*a(d) ) * a(n-k+1).
%e G.f.: A(x) = x - 2*x^2 - 2*x^3 + 10*x^4 + 14*x^5 - 86*x^6 - 126*x^7 + 858*x^8 + 1302*x^9 - 9378*x^10 - 14606*x^11 + ...
%t terms = 28; A[_] = 0; Do[A[x_] = x Exp[Sum[2 A[-x^(2 k - 1)]/(2 k - 1), {k, 1, terms}]] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
%t a[n_] := a[n] = SeriesCoefficient[x Product[((1 + x^k)/(1 - x^k))^((-1)^k a[k]), {k, 1, n - 1}], {x, 0, n}]; a[0] = 0; Table[a[n], {n, 0, 28}]
%Y Cf. A000081, A004111, A045648, A049075, A073075, A115593, A306768, A307365, A307366.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Apr 14 2019