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A116963 Inverse Moebius transform of the shifted tetrahedral numbers.

%I A116963 #33 Dec 30 2024 02:15:51
%S A116963 4,14,24,49,60,118,124,214,244,356,368,608,564,814,896,1183,1144,1668,
%T A116963 1544,2162,2168,2678,2604,3698,3336,4228,4304,5344,4964,6732,5988,
%U A116963 7728,7528,8924,8616,11297,9884,12214,12064,14668,13248,17132,15184,18928,18412,21038
%N A116963 Inverse Moebius transform of the shifted tetrahedral numbers.
%H A116963 Seiichi Manyama, <a href="/A116963/b116963.txt">Table of n, a(n) for n = 1..10000</a>
%F A116963 a(n) = Sum_{d|n} (d+1)*(d+2)*(d+3)/6 = Sum_{d|n} A000292(d+1).
%F A116963 G.f.: Sum_{k>0} (1/(1-x^k)^4 - 1). - _Seiichi Manyama_, Jun 12 2023
%F A116963 From _Amiram Eldar_, Dec 30 2024: (Start)
%F A116963 a(n) = (sigma_3(n) + 6*sigma_2(n) + 11*sigma_1(n) + 6*sigma_0(n))/6.
%F A116963 Dirichlet g.f.: zeta(s) * (zeta(s-3) + 6*zeta(s-2) + 11*zeta(s-1) + 6*zeta(s)) / 6.
%F A116963 Sum_{k=1..n} a(k) ~ (zeta(4)/24) * n^4. (End)
%e A116963 a(12) = ((1+1)*(1+2)*(1+3)/6) + ((2+1)*(2+2)*(2+3)/6) + ((3+1)*(3+2)*(3+3)/6) + ((4+1)*(4+2)*(4+3)/6) + ((6+1)*(6+2)*(6+3)/6) + ((12+1)*(12+2)*(12+3)/6) = 4 + 10 + 20 + 35 + 84 + 455 = 608.
%e A116963 a(13) = ((1+1)*(1+2)*(1+3)/6) + ((13+1)*(13+2)*(13+3)/6) = 4 + 560 = 564.
%t A116963 a[n_] := DivisorSum[n, Binomial[# + 3, 3] &]; Array[a, 50] (* _Amiram Eldar_, Jul 05 2023 *)
%o A116963 (PARI) my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, 1/(1-x^k)^4-1)) \\ _Seiichi Manyama_, Jun 12 2023
%Y A116963 See also: A007437 (inverse Moebius transform of triangular numbers).
%Y A116963 Cf. A000292, A007437, A007503.
%Y A116963 Cf. A013662, A059358, A363604, A363607, A363611.
%Y A116963 Cf. A000005, A000203, A001157, A001158.
%K A116963 easy,nonn
%O A116963 1,1
%A A116963 _Jonathan Vos Post_, Mar 31 2006