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%I #11 Oct 29 2020 03:24:14
%S 1,1,0,0,-3,1,-1,3,3,0,-12,15,-20,5,53,-113,180,-241,153,173,-652,787,
%T 628,-4801,11635,-18699,20775,-12315,-6109,21253,-7015,-61060,174382,
%U -260676,190623,130141,-549572,399845,1577502,-6670524,14603574,-21111528,16110192,14794188,-82586174
%N Expansion of Product_{k>=1} (1 + (x*(1 - x))^k).
%H Seiichi Manyama, <a href="/A307501/b307501.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d ) * (x*(1 - x))^k/k).
%F a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(k,n-k)*A000009(k).
%t nmax = 44; CoefficientList[Series[Product[(1 + (x (1 - x))^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t nmax = 44; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k/d + 1) d, {d, Divisors[k]}] (x (1 - x))^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
%t Table[Sum[(-1)^(n - k) Binomial[k, n - k] PartitionsQ[k], {k, 0, n}], {n, 0, 44}]
%Y Cf. A000009, A030528, A129519, A266108, A307496, A307500.
%K sign
%O 0,5
%A _Ilya Gutkovskiy_, Apr 11 2019