A290973 Write 2*x/(1-x) in the form Sum_{j>=1} ((1-x^j)^a(j) - 1).

-2, 1, 2, 3, 4, 6, 6, 10, 8, 15, 10, 25, 12, 28, 10, 60, 16, 25, 18, 125, 0, 66, 22, 218, 24, 91, -30, 420, 28, -387, 30, 2011, -88, 153, 28, -1894, 36, 190, -182, 8902, 40, -3234, 42, 2398, -132, 276, 46, 2340, 48, -2678, -510, 4641, 52, -1754, -198, 108400

Offset: 1

Author

Gus Wiseman, Aug 16 2017

Keywords

• sign

Examples

			2x/(1-x) = (1-x)^(-2) - 1 + (1-x^2)^1 - 1 + (1-x^3)^2 - 1 + (1-x^4)^3 - 1 + ...
		

Crossrefs

Cf. A048272, A220418, A260685, A281145, A289078, A289501, A290261, A290971.

Programs

Maple

a:= n-> add(binomial(n/d-1-a(d), n/d), d=
        numtheory[divisors](n) minus {n})-2:
seq(a(n), n=1..60);  # Alois P. Heinz, Aug 27 2017

Mathematica

nn=60;
rus=SolveAlways[Normal[Series[2x/(1-x)==Sum[(1-x^n)^a[n]-1,{n,nn}],{x,0,nn}]],x];
Array[a,nn]/.First[rus]

Formula

For all n > 0 we have: 2 = Sum_{d|n} binomial(-a(d) + n/d - 1, n/d).