cp's OEIS Frontend

A318581 Expansion of 1/(1 + x*Product_{k>=1} 1/(1 - x^k)).

%I A318581 #14 Jan 18 2020 11:09:39
%S A318581 1,-1,0,-1,0,-1,1,-1,3,-1,5,-2,7,-7,9,-16,11,-29,20,-46,45,-66,94,-95,
%T A318581 175,-161,294,-307,458,-594,715,-1096,1193,-1891,2132,-3106,3916,
%U A318581 -5063,7083,-8484,12347,-14770,20867,-26310,34898,-46771,58967,-81665,101680,-139951,178094,-237620
%N A318581 Expansion of 1/(1 + x*Product_{k>=1} 1/(1 - x^k)).
%H A318581 Seiichi Manyama, <a href="/A318581/b318581.txt">Table of n, a(n) for n = 0..5000</a>
%F A318581 G.f.: 1/(1 + x*Sum_{k>=0} A000041(k)*x^k).
%F A318581 a(0) = 1; a(n) = -Sum_{k=1..n} A000041(k-1)*a(n-k).
%e A318581 G.f. = 1 - x - x^3 - x^5 + x^6 - x^7 + 3*x^8 - x^9 + 5*x^10 - 2*x^11 + 7*x^12 - 7*x^13 + ...
%p A318581 seq(coeff(series((1+x*mul((1-x^k)^(-1),k=1..n))^(-1),x,n+1), x, n), n = 0 .. 55); # _Muniru A Asiru_, Aug 30 2018
%t A318581 nmax = 51; CoefficientList[Series[1/(1 + x Product[1/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
%t A318581 a[0] = 1; a[n_] := a[n] = -Sum[PartitionsP[k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 51}]
%Y A318581 Cf. similar sequences: A067687, A299105, A299106, A299208, A302017, A318582, A331484.
%Y A318581 Cf. A000041, A010815.
%K A318581 sign
%O A318581 0,9
%A A318581 _Ilya Gutkovskiy_, Aug 29 2018