A320900 - OEIS

%I #18 Jan 04 2025 05:36:15

%S 1,-2,7,-12,16,-17,29,-48,52,-42,67,-105,92,-79,142,-184,154,-143,191,

%T -262,266,-189,277,-441,341,-262,430,-495,436,-402,497,-712,634,-444,

%U 674,-897,704,-553,878,-1118,862,-766,947,-1189,1222,-807,1129,-1753,1254,-992

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

%H Seiichi Manyama, <a href="/A320900/b320900.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

%F a(n) = (A002129(n) + A321543(n)) / 2. - _Amiram Eldar_, Jan 04 2025

%p 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

%t nmax = 50; Rest[CoefficientList[Series[Sum[x^k/(1 + x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x]]

%t Table[Sum[(-1)^(d + 1) d (d + 1)/2, {d, Divisors[n]}], {n, 50}]

%o (PARI) a(n) = sumdiv(n, d, (-1)^(d+1)*d*(d + 1)/2); \\ _Amiram Eldar_, Jan 04 2025

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

%Y Cf. A363022, A363615, A363630.

%K sign,look

%O 1,2

%A _Ilya Gutkovskiy_, Oct 23 2018