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

%I A363616 #30 Jan 04 2025 05:46:27
%S A363616 0,0,0,1,-4,10,-20,36,-56,80,-120,176,-220,266,-368,491,-560,634,-816,
%T A363616 1050,-1160,1210,-1540,1982,-2028,2080,-2656,3192,-3276,3380,-4060,
%U A363616 4986,-5080,4896,-6008,7345,-7140,6954,-8656,10224,-9880,9796,-11480,13552,-13668,12650
%N A363616 Expansion of Sum_{k>0} x^(4*k)/(1+x^k)^4.
%H A363616 Seiichi Manyama, <a href="/A363616/b363616.txt">Table of n, a(n) for n = 1..10000</a>
%F A363616 G.f.: Sum_{k>0} binomial(k-1,3) * (-x)^k/(1 - x^k).
%F A363616 a(n) = Sum_{d|n} (-1)^d * binomial(d-1,3).
%F A363616 a(n) = A128315(n, 4), for n >= 4. - _G. C. Greubel_, Jun 22 2024
%F A363616 a(n) = -(A138503(n) - 6*A321543(n) + 11*A002129(n) - 6*A048272(n)) / 6. - _Amiram Eldar_, Jan 04 2025
%t A363616 a[n_] := DivisorSum[n, (-1)^# * Binomial[# - 1, 3] &]; Array[a, 50] (* _Amiram Eldar_, Jul 25 2023 *)
%o A363616 (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)))
%o A363616 (PARI) a(n) = sumdiv(n, d, (-1)^d*binomial(d-1, 3));
%o A363616 (Magma)
%o A363616 A363616:= func< n | (&+[(-1)^d*Binomial(d-1,3): d in Divisors(n)]) >;
%o A363616 [A363616(n): n in [1..60]]; // _G. C. Greubel_, Jun 22 2024
%o A363616 (SageMath)
%o A363616 def A363616(n): return sum(0^(n%j)*(-1)^j*binomial(j-1,3) for j in range(4, n+1))
%o A363616 [A363616(n) for n in range(1,61)] # _G. C. Greubel_, Jun 22 2024
%Y A363616 Cf. A002129, A128315, A138503, A321543, A325940, A363611, A363615.
%K A363616 sign
%O A363616 1,5
%A A363616 _Seiichi Manyama_, Jun 11 2023