A307500 -id:A307500 - cp's OEIS frontend

A307501 Expansion of Product_{k>=1} (1 + (x*(1 - x))^k).

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

Offset: 0

Ilya Gutkovskiy, Apr 11 2019

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000009, A030528, A129519, A266108, A307496, A307500.

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}]

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).

A327671 Expansion of Product_{k>=1} (1 - (x*(1 - x))^k).

Original entry on oeis.org

1, -1, 0, 2, -1, 1, -5, 11, -17, 26, -36, 35, -22, 19, -67, 219, -480, 687, -469, -573, 2508, -4785, 6370, -6445, 5235, -4543, 8681, -26815, 75043, -173159, 334721, -563200, 876876, -1363232, 2208921, -3621971, 5631540, -7897299, 9738858, -10479294, 9989646, -9350820

Offset: 0

Seiichi Manyama, Sep 21 2019

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Convolution inverse of A307500.

Cf. A307310, A307501.

m = 41; CoefficientList[Series[Product[(1 - (x*(1 - x))^k), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 07 2021 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, 1-(x*(1-x))^k))

N=66; x='x+O('x^N); Vec(exp(-sum(k=1, N, sigma(k)*(x*(1-x))^k/k)))

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

cp's OEIS Frontend

A307501 Expansion of Product_{k>=1} (1 + (x*(1 - x))^k).

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