A363606 - OEIS

%I #22 Dec 30 2024 02:16:29

%S 0,1,6,22,56,133,252,484,798,1344,2002,3157,4368,6441,8630,12112,

%T 15504,21274,26334,35014,42762,55133,65780,84349,98336,123124,143304,

%U 176373,201376,247380,278256,336744,379000,451402,502250,600055,658008,775733,855042

%N Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^6.

%H Seiichi Manyama, <a href="/A363606/b363606.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f.: Sum_{k>0} binomial(k+3,5) * x^k/(1 - x^k).

%F a(n) = Sum_{d|n} binomial(d+3,5).

%F From _Amiram Eldar_, Dec 30 2024: (Start)

%F a(n) = (sigma_5(n) + 5*sigma_4(n) + 5*sigma_3(n) - 5*sigma_2(n) - 6*sigma_1(n)) / 120.

%F Dirichlet g.f.: zeta(s) * (zeta(s-5) + 5*zeta(s-4) + 5*zeta(s-3) - 5*zeta(s-2) - 6*zeta(s-1)) / 120.

%F Sum_{k=1..n} a(k) ~ (zeta(6)/720) * n^6. (End)

%t a[n_] := DivisorSum[n, Binomial[# + 3, 5] &]; Array[a, 40] (* _Amiram Eldar_, Jul 25 2023 *)

%o (PARI) my(N=40, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^k)^6)))

%o (PARI) a(n) = my(f = factor(n)); (sigma(f, 5) + 5*sigma(f, 4) + 5*sigma(f, 3) - 5*sigma(f, 2) - 6*sigma(f)) / 120; \\ _Amiram Eldar_, Dec 30 2024

%Y Cf. A013664, A032741, A065608, A069153, A363604, A363605.

%Y Cf. A000203, A001157, A001158, A001159, A001160.

%K nonn,easy

%O 1,3

%A _Seiichi Manyama_, Jun 11 2023