cp's OEIS Frontend

A363610 Expansion of Sum_{k>0} x^(3*k)/(1-x^k)^3.

%I A363610 #24 Jan 02 2025 09:35:48
%S A363610 0,0,1,3,6,11,15,24,29,42,45,69,66,93,98,129,120,175,153,216,206,255,
%T A363610 231,343,282,366,354,447,378,550,435,594,542,648,582,828,630,819,770,
%U A363610 978,780,1114,861,1161,1072,1221,1035,1529,1143,1494,1346,1644,1326,1878,1482,1953
%N A363610 Expansion of Sum_{k>0} x^(3*k)/(1-x^k)^3.
%H A363610 Seiichi Manyama, <a href="/A363610/b363610.txt">Table of n, a(n) for n = 1..10000</a>
%F A363610 G.f.: Sum_{k>0} binomial(k-1,2) * x^k/(1 - x^k).
%F A363610 a(n) = Sum_{d|n} binomial(d-1,2).
%F A363610 From _Amiram Eldar_, Jan 01 2025: (Start)
%F A363610 a(n) = (sigma_2(n) - 3*sigma_1(n) + 2*sigma_0(n)) / 2.
%F A363610 Dirichlet g.f.: zeta(s) * (zeta(s-2) - 3*zeta(s-1) + 2*zeta(s)) / 2.
%F A363610 Sum_{k=1..n} a(k) ~ (zeta(3)/6) * n^3. (End)
%t A363610 a[n_] := DivisorSum[n, Binomial[# - 1, 2] &]; Array[a, 50] (* _Amiram Eldar_, Jul 25 2023 *)
%o A363610 (PARI) my(N=60, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(3*k)/(1-x^k)^3)))
%o A363610 (PARI) a(n) = my(f = factor(n)); (sigma(f, 2) - 3*sigma(f) + 2*numdiv(f)) / 2; \\ _Amiram Eldar_, Jan 01 2025
%Y A363610 Cf. A000005, A000203, A001157, A002117, A065608, A363611.
%Y A363610 Cf. A007437, A069153.
%K A363610 nonn,easy
%O A363610 1,4
%A A363610 _Seiichi Manyama_, Jun 11 2023