login
G.f. A(x) satisfies: (1 - x) = A(x)*A(x^2)^2*A(x^3)^3*A(x^4)^4* ... *A(x^k)^k* ...
0

%I #5 Apr 21 2019 07:50:24

%S 1,-1,2,1,0,8,-7,22,-6,13,29,-11,82,-36,114,13,103,88,88,275,-20,549,

%T -200,1007,-144,811,730,188,2093,-777,3538,-643,4083,-537,4562,2478,

%U 1973,8062,-3508,17362,-8164,20281,-2227,17483,8605,2946,30190,-6085,53176,-28913,78516

%N G.f. A(x) satisfies: (1 - x) = A(x)*A(x^2)^2*A(x^3)^3*A(x^4)^4* ... *A(x^k)^k* ...

%C Convolution inverse of A307648.

%F G.f.: Product_{k>=1} (1 - x^k)^(mu(k)*k).

%F G.f.: exp(-Sum_{k>=1} A046970(k)*x^k/k).

%e G.f.: A(x) = 1 - x + 2*x^2 + x^3 + 8*x^5 - 7*x^6 + 22 x^7 - 6*x^8 + 13*x^9 + 29*x^10 - 11*x^11 + 82*x^12 - 36*x^13 + ...

%t terms = 50; CoefficientList[Series[Product[(1 - x^k)^(MoebiusMu[k] k), {k, 1, terms}], {x, 0, terms}], x]

%t terms = 50; CoefficientList[Series[Exp[-Sum[Sum[MoebiusMu[d] d^2, {d, Divisors[k]}] x^k/k, {k, 1, terms}]], {x, 0, terms}], x]

%t terms = 50; A[_] = 1; Do[A[x_] = (1 - x)/Product[A[x^k]^k, {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

%Y Cf. A008683, A046970, A055615, A117208, A307648.

%K sign

%O 0,3

%A _Ilya Gutkovskiy_, Apr 20 2019