A300786 L.g.f.: log(Product_{k>=1} (1 + k*x^k)) = Sum_{n>=1} a(n)*x^n/n.

1, 3, 10, 7, 26, 24, 50, -33, 163, 38, 122, -188, 170, 108, 1580, -1793, 290, -273, 362, -1678, 9404, 3248, 530, -49092, 16251, 14862, 66340, 14000, 842, -135556, 962, -429057, 547172, 258386, 509500, -1392821, 1370, 1043160, 4813052, -8088838, 1682, -9267612, 1850, 8218844, 53396438

Offset

1,2

Links

Table of n, a(n) for n=1..45.

Formula

L.g.f.: Sum_{j>=1} Sum_{k>=1} (-1)^(j+1)*k^j*x^(j*k)/j = Sum_{n>=1} a(n)*x^n/n.

G.f.: Sum_{k>=1} k^2*x^k/(1 + k*x^k).

a(n) = Sum_{d|n} (-d)^(n/d+1).

Example

L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 7*x^4/4 + 26*x^5/5 + 24*x^6/6 + 50*x^7/7 - 33*x^8/8 + 163*x^9/9 + 38*x^10/10 + ...
exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 7*x^4 + 15*x^5 + 25*x^6 + 43*x^7 + 64*x^8 + 120*x^9 + 186*x^10 + ... + A022629(n)*x^n + ...

Mathematica

nmax = 45; Rest[CoefficientList[Series[Log[Product[(1 + k x^k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 45; Rest[CoefficientList[Series[Sum[Sum[(-1)^(j + 1) k^j x^(j k)/j, {k, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 45; Rest[CoefficientList[Series[Sum[k^2 x^k/(1 + k x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
a[n_] := Sum[(-d)^(n/d + 1), {d, Divisors[n]}]; Table[a[n], {n, 1, 45}]

Crossrefs

Cf. A022629, A076717, A078308.

Sequence in context: A261836 A301937 A185139 * A182241 A033152 A281178

Adjacent sequences: A300783 A300784 A300785 * A300787 A300788 A300789

Keyword

sign

Author

Ilya Gutkovskiy, Mar 12 2018

Status

approved