A307500 - OEIS

%I #11 Oct 31 2020 02:53:54

%S 1,1,1,-1,-2,-4,3,-1,17,-16,21,-57,67,-130,305,-536,995,-1726,2652,

%T -4286,7320,-13043,24458,-45405,81415,-141724,239755,-400603,676872,

%U -1171076,2072334,-3695550,6519951,-11279015,19188230,-32462795,55334284,-95718737,167673672,-294894076

%N Expansion of Product_{k>=1} 1/(1 - (x*(1 - x))^k).

%H Seiichi Manyama, <a href="/A307500/b307500.txt">Table of n, a(n) for n = 0..1000</a>

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

%F a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(k,n-k)*A000041(k).

%t nmax = 39; CoefficientList[Series[Product[1/(1 - (x (1 - x))^k), {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 39; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k] (x (1 - x))^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]

%t Table[Sum[(-1)^(n - k) Binomial[k, n - k] PartitionsP[k], {k, 0, n}], {n, 0, 39}]

%Y Cf. A000041, A030528, A218482, A238441, A286955, A307501.

%K sign

%O 0,5

%A _Ilya Gutkovskiy_, Apr 11 2019