%I #30 Jul 05 2023 03:00:23

%S 4,14,24,49,60,118,124,214,244,356,368,608,564,814,896,1183,1144,1668,

%T 1544,2162,2168,2678,2604,3698,3336,4228,4304,5344,4964,6732,5988,

%U 7728,7528,8924,8616,11297,9884,12214,12064,14668,13248,17132,15184,18928,18412,21038

%N Inverse Moebius transform of the shifted tetrahedral numbers.

%H Seiichi Manyama, <a href="/A116963/b116963.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{d|n} (d+1)*(d+2)*(d+3)/6 = Sum_{d|n} A000292(d+1).

%F G.f.: Sum_{k>0} (1/(1-x^k)^4 - 1). - _Seiichi Manyama_, Jun 12 2023

%e 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 a(13) = ((1+1)*(1+2)*(1+3)/6) + ((13+1)*(13+2)*(13+3)/6) = 4 + 560 = 564.

%t a[n_] := DivisorSum[n, Binomial[# + 3, 3] &]; Array[a, 50] (* _Amiram Eldar_, Jul 05 2023 *)

%o (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 See also: A007437 (inverse Moebius transform of triangular numbers).

%Y Cf. A000292, A007437, A007503.

%Y Cf. A059358, A363604, A363607, A363611.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Mar 31 2006