A307656 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* ...

1, -1, 2, 1, 0, 8, -7, 22, -6, 13, 29, -11, 82, -36, 114, 13, 103, 88, 88, 275, -20, 549, -200, 1007, -144, 811, 730, 188, 2093, -777, 3538, -643, 4083, -537, 4562, 2478, 1973, 8062, -3508, 17362, -8164, 20281, -2227, 17483, 8605, 2946, 30190, -6085, 53176, -28913, 78516

Offset: 0

Author

Ilya Gutkovskiy, Apr 20 2019

Keywords

• sign

Comments

Convolution inverse of A307648.

Examples

			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 + ...
		

Crossrefs

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

Programs

Mathematica

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

Formula

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

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