A320901 Expansion of Sum_{k>=1} x^k/(1 + x^k)^4.

1, -3, 11, -23, 36, -49, 85, -143, 176, -188, 287, -433, 456, -479, 726, -959, 970, -1024, 1331, -1748, 1866, -1741, 2301, -3153, 2961, -2824, 3830, -4559, 4496, -4514, 5457, -6943, 6842, -6174, 7890, -9844, 9140, -8553, 11126, -13348, 12342, -11998, 14191, -16941

Offset: 1

Author

Ilya Gutkovskiy, Oct 23 2018

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A000292, A000593, A001157, A001158, A002129, A050999, A051000, A059358, A138503, A320900, A321543.

Cf. A363598, A363616, A363617, A363631.

Programs

Maple

seq(coeff(series(add(x^k/(1+x^k)^4,k=1..n),x,n+1), x, n), n = 1 .. 45); # Muniru A Asiru, Oct 23 2018

Mathematica

nmax = 44; Rest[CoefficientList[Series[Sum[x^k/(1 + x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[(-1)^(d + 1) d (d + 1) (d + 2)/6, {d, Divisors[n]}], {n, 44}]

PARI

a(n) = sumdiv(n, d, (-1)^(d+1)*d*(d + 1)*(d + 2)/6); \\ Amiram Eldar, Jan 04 2025

Formula

G.f.: Sum_{k>=1} (-1)^(k+1)*A000292(k)*x^k/(1 - x^k).

a(n) = Sum_{d|n} (-1)^(d+1)*d*(d + 1)*(d + 2)/6.

a(n) = (4*A000593(n) + 6*A050999(n) + 2*A051000(n) - 2*A000203(n) - 3*A001157(n) - A001158(n))/6.

a(n) = (A138503(n) + 3*A321543(n) + 2*A002129(n)) / 6. - Amiram Eldar, Jan 04 2025