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A363605 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^5.

%I A363605 #19 Dec 30 2024 02:16:22
%S A363605 0,1,5,16,35,76,126,226,335,531,715,1092,1365,1947,2420,3286,3876,
%T A363605 5251,5985,7861,8986,11342,12650,16252,17585,21841,24086,29367,31465,
%U A363605 38946,40920,49662,53080,62782,66206,80082,82251,97376,102640,120001,123410,146628
%N A363605 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^5.
%H A363605 Seiichi Manyama, <a href="/A363605/b363605.txt">Table of n, a(n) for n = 1..10000</a>
%F A363605 G.f.: Sum_{k>0} binomial(k+2,4) * x^k/(1 - x^k).
%F A363605 a(n) = Sum_{d|n} binomial(d+2,4).
%F A363605 From _Amiram Eldar_, Dec 30 2024: (Start)
%F A363605 a(n) = (sigma_4(n) + 2*sigma_3(n) - sigma_2(n) - 2*sigma_1(n)) / 24.
%F A363605 Dirichlet g.f.: zeta(s) * (zeta(s-4) + 2*zeta(s-3) - zeta(s-2) - 2*zeta(s-1)) / 24.
%F A363605 Sum_{k=1..n} a(k) ~ (zeta(5)/120) * n^5. (End)
%t A363605 a[n_] := DivisorSum[n, Binomial[# + 2, 4] &]; Array[a, 40] (* _Amiram Eldar_, Jul 25 2023 *)
%o A363605 (PARI) my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^k)^5)))
%o A363605 (PARI) a(n) = my(f = factor(n)); (sigma(f, 4) + 2*sigma(f, 3) - sigma(f, 2) - 2*sigma(f)) / 24; \\ _Amiram Eldar_, Dec 30 2024
%Y A363605 Cf. A013663, A032741, A065608, A069153, A363604, A363606.
%Y A363605 Cf. A000203, A001157, A001158, A001159.
%K A363605 nonn,easy
%O A363605 1,3
%A A363605 _Seiichi Manyama_, Jun 11 2023