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

1,1

Links

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

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

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

Cf. A000292, A007437, A007503.

Cf. A059358, A363604, A363607, A363611.

Sequence in context: A277591 A017317 A195973 * A094930 A289665 A080286

Adjacent sequences: A116960 A116961 A116962 * A116964 A116965 A116966

Keyword

easy,nonn

Author

Jonathan Vos Post, Mar 31 2006

Status

approved