A363615 Expansion of Sum_{k>0} x^(3*k)/(1+x^k)^3.

0, 0, 1, -3, 6, -9, 15, -24, 29, -30, 45, -67, 66, -63, 98, -129, 120, -117, 153, -204, 206, -165, 231, -341, 282, -234, 354, -417, 378, -354, 435, -594, 542, -408, 582, -770, 630, -513, 770, -966, 780, -702, 861, -1071, 1072, -759, 1035, -1527, 1143, -930, 1346

Offset: 1

Author

Seiichi Manyama, Jun 11 2023

Keywords

• sign

Links

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

Crossrefs

Cf. A002129, A048272, A128315, A325940, A321543, A363610, A363616.

Programs

Magma

A363615:= func< n | -(&+[(-1)^d*Binomial(d-1,2): d in Divisors(n)]) >;
[A363615(n): n in [1..60]]; // G. C. Greubel, Jun 22 2024

Mathematica

a[n_] := -DivisorSum[n, (-1)^#*Binomial[# - 1, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 18 2023 *)

PARI

my(N=60, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(3*k)/(1+x^k)^3)))

PARI

a(n) = -sumdiv(n, d, (-1)^d*binomial(d-1, 2));

SageMath

def A363615(n): return sum(0^(n%j)*(-1)^(j+1)*binomial(j-1,2) for j in range(1, n+1))
[A363615(n) for n in range(1,61)] # G. C. Greubel, Jun 22 2024

Formula

G.f.: -Sum_{k>0} binomial(k-1,2) * (-x)^k/(1 - x^k).

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

a(n) = A128315(n, 3), for n >= 3. - G. C. Greubel, Jun 22 2024

a(n) = (A321543(n) - 3*A002129(n) + 2*A048272(n)) / 2. - Amiram Eldar, Jan 04 2025