A307365 - OEIS

%I #7 Apr 05 2019 17:46:33

%S 0,1,-1,-1,2,1,-4,-3,11,10,-36,-32,122,105,-420,-368,1497,1336,-5491,

%T -4919,20477,18393,-77397,-69883,296306,268711,-1146538,-1042924,

%U 4475265,4081598,-17600475,-16091719,69681964,63845971,-277494594,-254730047,1110782803,1021361912

%N G.f. A(x) satisfies: A(x) = x*exp(A(-x) + A(-x^2)/2 + A(-x^3)/3 + A(-x^4)/4 + ...).

%F G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} 1/(1 - x^n)^((-1)^n*a(n)).

%F Recurrence: a(n+1) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^d*d*a(d) ) * a(n-k+1).

%e G.f.: A(x) = x - x^2 - x^3 + 2*x^4 + x^5 - 4*x^6 - 3*x^7 + 11*x^8 + 10*x^9 - 36*x^10 - 32*x^11 + ...

%t terms = 37; A[_] = 0; Do[A[x_] = x Exp[Sum[A[-x^k]/k, {k, 1, terms}]] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

%t a[n_] := a[n] = SeriesCoefficient[x Product[1/(1 - x^k)^((-1)^k a[k]), {k, 1, n - 1}], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 0, 37}]

%Y Cf. A000081, A004111, A045648, A049075, A307366.

%K sign

%O 0,5

%A _Ilya Gutkovskiy_, Apr 05 2019