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Expansion of Sum_{k>0} x^(4*k)/(1-x^k)^4.
6

%I #33 Jan 02 2025 09:35:59

%S 0,0,0,1,4,10,20,36,56,88,120,176,220,306,368,491,560,746,816,1058,

%T 1160,1450,1540,1982,2028,2520,2656,3232,3276,4116,4060,4986,5080,

%U 6016,6008,7457,7140,8586,8656,10232,9880,12116,11480,13792,13668,15730,15180,18652,17316,20536

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

%H Seiichi Manyama, <a href="/A363611/b363611.txt">Table of n, a(n) for n = 1..10000</a>

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

%F a(n) = Sum_{d|n} binomial(d-1,3).

%F From _Amiram Eldar_, Jan 01 2025: (Start)

%F a(n) = (sigma_3(n) - 6*sigma_2(n) + 11*sigma_1(n) - 6*sigma_0(n)) / 6.

%F Dirichlet g.f.: zeta(s) * (zeta(s-3) - 6*zeta(s-2) + 11*zeta(s-1) - 6*zeta(s)) / 6.

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

%t a[n_] := DivisorSum[n, Binomial[# - 1, 3] &]; Array[a, 50] (* _Amiram Eldar_, Jul 25 2023 *)

%o (PARI) my(N=60, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/(1-x^k)^4)))

%o (PARI) a(n) = my(f = factor(n)); (sigma(f, 3) - 6*sigma(f, 2) + 11*sigma(f) - 6*numdiv(f)) / 6; \\ _Amiram Eldar_, Jan 01 2025

%Y Cf. A000005, A000203, A001157, A001158, A013662, A065608, A363610.

%Y Cf. A059358, A363604, A363607.

%K nonn,easy

%O 1,5

%A _Seiichi Manyama_, Jun 11 2023