cp's OEIS Frontend

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

%I A290973 #8 Aug 27 2017 21:47:42
%S A290973 -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,
%T A290973 91,-30,420,28,-387,30,2011,-88,153,28,-1894,36,190,-182,8902,40,
%U A290973 -3234,42,2398,-132,276,46,2340,48,-2678,-510,4641,52,-1754,-198,108400
%N A290973 Write 2*x/(1-x) in the form Sum_{j>=1} ((1-x^j)^a(j) - 1).
%F A290973 For all n > 0 we have: 2 = Sum_{d|n} binomial(-a(d) + n/d - 1, n/d).
%e A290973 2x/(1-x) = (1-x)^(-2) - 1 + (1-x^2)^1 - 1 + (1-x^3)^2 - 1 + (1-x^4)^3 - 1 + ...
%p A290973 a:= n-> add(binomial(n/d-1-a(d), n/d), d=
%p A290973         numtheory[divisors](n) minus {n})-2:
%p A290973 seq(a(n), n=1..60);  # _Alois P. Heinz_, Aug 27 2017
%t A290973 nn=60;
%t A290973 rus=SolveAlways[Normal[Series[2x/(1-x)==Sum[(1-x^n)^a[n]-1,{n,nn}],{x,0,nn}]],x];
%t A290973 Array[a,nn]/.First[rus]
%Y A290973 Cf. A048272, A220418, A260685, A281145, A289078, A289501, A290261, A290971.
%K A290973 sign
%O A290973 1,1
%A A290973 _Gus Wiseman_, Aug 16 2017