A320900 Expansion of Sum_{k>=1} x^k/(1 + x^k)^3.

1, -2, 7, -12, 16, -17, 29, -48, 52, -42, 67, -105, 92, -79, 142, -184, 154, -143, 191, -262, 266, -189, 277, -441, 341, -262, 430, -495, 436, -402, 497, -712, 634, -444, 674, -897, 704, -553, 878, -1118, 862, -766, 947, -1189, 1222, -807, 1129, -1753, 1254, -992

Offset: 1

Author

Ilya Gutkovskiy, Oct 23 2018

Keywords

• sign
• look

Links

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

Crossrefs

Cf. A000203, A000217, A000593, A001157, A002129, A007437, A050999, A064027, A320901, A321543.

Cf. A363022, A363615, A363630.

Programs

Maple

seq(coeff(series(add(x^k/(1+x^k)^3,k=1..n),x,n+1), x, n), n = 1 .. 50); # Muniru A Asiru, Oct 23 2018

Mathematica

nmax = 50; Rest[CoefficientList[Series[Sum[x^k/(1 + x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[(-1)^(d + 1) d (d + 1)/2, {d, Divisors[n]}], {n, 50}]

PARI

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

Formula

G.f.: Sum_{k>=1} (-1)^(k+1)*A000217(k)*x^k/(1 - x^k).

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

a(n) = A000593(n) + A050999(n) - (A000203(n) + A001157(n))/2.

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