A116963 Inverse Moebius transform of the shifted tetrahedral numbers.

4, 14, 24, 49, 60, 118, 124, 214, 244, 356, 368, 608, 564, 814, 896, 1183, 1144, 1668, 1544, 2162, 2168, 2678, 2604, 3698, 3336, 4228, 4304, 5344, 4964, 6732, 5988, 7728, 7528, 8924, 8616, 11297, 9884, 12214, 12064, 14668, 13248, 17132, 15184, 18928, 18412, 21038

Offset: 1

Author

Jonathan Vos Post, Mar 31 2006

Keywords

• easy
• nonn

Examples

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

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

Cf. A000292, A007437, A007503.

Cf. A013662, A059358, A363604, A363607, A363611.

Cf. A000005, A000203, A001157, A001158.

Programs

Mathematica

a[n_] := DivisorSum[n, Binomial[# + 3, 3] &]; Array[a, 50] (* Amiram Eldar, Jul 05 2023 *)

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

Formula

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

G.f.: Sum_{k>0} (1/(1-x^k)^4 - 1). - Seiichi Manyama, Jun 12 2023

From Amiram Eldar, Dec 30 2024: (Start)

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

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

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