A363607 - OEIS

%I #25 Dec 30 2024 02:16:36

%S 0,0,1,4,10,21,35,60,85,130,165,245,286,399,466,620,680,921,969,1274,

%T 1366,1705,1771,2325,2310,2886,3010,3679,3654,4666,4495,5580,5622,

%U 6664,6590,8285,7770,9405,9426,11210,10660,13230,12341,14953,14740,16951,16215,20181

%N Expansion of Sum_{k>0} x^(3*k)/(1-x^k)^4.

%H Seiichi Manyama, <a href="/A363607/b363607.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

%F a(n) = (sigma_3(n) - 3*sigma_2(n) + 2*sigma_1(n)) / 6.

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

%F Sum_{k=1..n} a(k) ~ (zeta(4)/24) * n^4. (End)

%t a[n_] := DivisorSum[n, Binomial[#, 3] &]; Array[a, 50] (* _Amiram Eldar_, Jul 25 2023 *)

%o (PARI) my(N=50, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(3*k)/(1-x^k)^4)))

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

%Y Cf. A013662, A069153, A363608.

%Y Cf. A059358, A363604, A363611.

%Y Cf. A000203, A001157, A001158.

%K nonn,easy,changed

%O 1,4

%A _Seiichi Manyama_, Jun 11 2023