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A363608 Expansion of Sum_{k>0} x^(4*k)/(1-x^k)^5.

%I A363608 #22 Dec 30 2024 02:16:45
%S A363608 0,0,0,1,5,15,35,71,126,215,330,511,715,1036,1370,1891,2380,3201,3876,
%T A363608 5061,6020,7645,8855,11207,12655,15665,17676,21512,23751,29000,31465,
%U A363608 37851,41250,48756,52400,62602,66045,77691,82966,96521,101270,118966,123410,143397
%N A363608 Expansion of Sum_{k>0} x^(4*k)/(1-x^k)^5.
%H A363608 Seiichi Manyama, <a href="/A363608/b363608.txt">Table of n, a(n) for n = 1..10000</a>
%F A363608 G.f.: Sum_{k>0} binomial(k,4) * x^k/(1 - x^k).
%F A363608 a(n) = Sum_{d|n} binomial(d,4).
%F A363608 From _Amiram Eldar_, Dec 30 2024: (Start)
%F A363608 a(n) = (sigma_4(n) - 6*sigma_3(n) + 11*sigma_2(n) - 6*sigma_1(n)) / 24.
%F A363608 Dirichlet g.f.: zeta(s) * (zeta(s-4) - 6*zeta(s-3) + 11*zeta(s-2) - 6*zeta(s-1)) / 24.
%F A363608 Sum_{k=1..n} a(k) ~ (zeta(5)/120) * n^5. (End)
%t A363608 a[n_] := DivisorSum[n, Binomial[#, 4] &]; Array[a, 50] (* _Amiram Eldar_, Jul 25 2023 *)
%o A363608 (PARI) my(N=50, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/(1-x^k)^5)))
%o A363608 (PARI) a(n) = my(f = factor(n)); (sigma(f, 4) - 6*sigma(f, 3) + 11*sigma(f, 2) - 6*sigma(f)) / 24; \\ _Amiram Eldar_, Dec 30 2024
%Y A363608 Cf. A013663, A069153, A363607.
%Y A363608 Cf. A000203, A001157, A001158, A001159.
%K A363608 nonn,easy
%O A363608 1,5
%A A363608 _Seiichi Manyama_, Jun 11 2023