A363616 Expansion of Sum_{k>0} x^(4*k)/(1+x^k)^4.

0, 0, 0, 1, -4, 10, -20, 36, -56, 80, -120, 176, -220, 266, -368, 491, -560, 634, -816, 1050, -1160, 1210, -1540, 1982, -2028, 2080, -2656, 3192, -3276, 3380, -4060, 4986, -5080, 4896, -6008, 7345, -7140, 6954, -8656, 10224, -9880, 9796, -11480, 13552, -13668, 12650

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, A128315, A138503, A321543, A325940, A363611, A363615.

Programs

Magma

A363616:= func< n | (&+[(-1)^d*Binomial(d-1,3): d in Divisors(n)]) >;
[A363616(n): n in [1..60]]; // G. C. Greubel, Jun 22 2024

Mathematica

a[n_] := DivisorSum[n, (-1)^# * Binomial[# - 1, 3] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)

PARI

my(N=50, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/(1+x^k)^4)))

PARI

a(n) = sumdiv(n, d, (-1)^d*binomial(d-1, 3));

SageMath

def A363616(n): return sum(0^(n%j)*(-1)^j*binomial(j-1,3) for j in range(4, n+1))
[A363616(n) for n in range(1,61)] # G. C. Greubel, Jun 22 2024

Formula

G.f.: Sum_{k>0} binomial(k-1,3) * (-x)^k/(1 - x^k).

a(n) = Sum_{d|n} (-1)^d * binomial(d-1,3).

a(n) = A128315(n, 4), for n >= 4. - G. C. Greubel, Jun 22 2024

a(n) = -(A138503(n) - 6*A321543(n) + 11*A002129(n) - 6*A048272(n)) / 6. - Amiram Eldar, Jan 04 2025