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 A307501 Expansion of Product_{k>=1} (1 + (x*(1 - x))^k). 3
 1, 1, 0, 0, -3, 1, -1, 3, 3, 0, -12, 15, -20, 5, 53, -113, 180, -241, 153, 173, -652, 787, 628, -4801, 11635, -18699, 20775, -12315, -6109, 21253, -7015, -61060, 174382, -260676, 190623, 130141, -549572, 399845, 1577502, -6670524, 14603574, -21111528, 16110192, 14794188, -82586174 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS FORMULA G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d ) * (x*(1 - x))^k/k). a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(k,n-k)*A000009(k). MATHEMATICA nmax = 44; CoefficientList[Series[Product[(1 + (x (1 - x))^k), {k, 1, nmax}], {x, 0, nmax}], x] 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] Table[Sum[(-1)^(n - k) Binomial[k, n - k] PartitionsQ[k], {k, 0, n}], {n, 0, 44}] CROSSREFS Cf. A000009, A030528, A266108, A307496, A307500. Sequence in context: A324079 A184831 A033989 * A169941 A099545 A300867 Adjacent sequences:  A307498 A307499 A307500 * A307502 A307503 A307504 KEYWORD sign AUTHOR Ilya Gutkovskiy, Apr 11 2019 STATUS approved

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Last modified February 19 13:03 EST 2020. Contains 332044 sequences. (Running on oeis4.)