cp's OEIS Frontend

A363611 Expansion of Sum_{k>0} x^(4*k)/(1-x^k)^4.

%I A363611 #33 Jan 02 2025 09:35:59
%S A363611 0,0,0,1,4,10,20,36,56,88,120,176,220,306,368,491,560,746,816,1058,
%T A363611 1160,1450,1540,1982,2028,2520,2656,3232,3276,4116,4060,4986,5080,
%U A363611 6016,6008,7457,7140,8586,8656,10232,9880,12116,11480,13792,13668,15730,15180,18652,17316,20536
%N A363611 Expansion of Sum_{k>0} x^(4*k)/(1-x^k)^4.
%H A363611 Seiichi Manyama, <a href="/A363611/b363611.txt">Table of n, a(n) for n = 1..10000</a>
%F A363611 G.f.: Sum_{k>0} binomial(k-1,3) * x^k/(1 - x^k).
%F A363611 a(n) = Sum_{d|n} binomial(d-1,3).
%F A363611 From _Amiram Eldar_, Jan 01 2025: (Start)
%F A363611 a(n) = (sigma_3(n) - 6*sigma_2(n) + 11*sigma_1(n) - 6*sigma_0(n)) / 6.
%F A363611 Dirichlet g.f.: zeta(s) * (zeta(s-3) - 6*zeta(s-2) + 11*zeta(s-1) - 6*zeta(s)) / 6.
%F A363611 Sum_{k=1..n} a(k) ~ (zeta(4)/24) * n^4. (End)
%t A363611 a[n_] := DivisorSum[n, Binomial[# - 1, 3] &]; Array[a, 50] (* _Amiram Eldar_, Jul 25 2023 *)
%o A363611 (PARI) my(N=60, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/(1-x^k)^4)))
%o A363611 (PARI) a(n) = my(f = factor(n)); (sigma(f, 3) - 6*sigma(f, 2) + 11*sigma(f) - 6*numdiv(f)) / 6; \\ _Amiram Eldar_, Jan 01 2025
%Y A363611 Cf. A000005, A000203, A001157, A001158, A013662, A065608, A363610.
%Y A363611 Cf. A059358, A363604, A363607.
%K A363611 nonn,easy
%O A363611 1,5
%A A363611 _Seiichi Manyama_, Jun 11 2023