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

0,5

Links

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

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, A129519, 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