A059358 Coefficients in expansion of Sum_{n >= 1} x^n/(1-x^n)^4.

0, 1, 5, 11, 25, 36, 71, 85, 145, 176, 260, 287, 455, 456, 649, 726, 961, 970, 1376, 1331, 1820, 1866, 2315, 2301, 3175, 2961, 3736, 3830, 4729, 4496, 5966, 5457, 6945, 6842, 8114, 7890, 10196, 9140, 11215, 11126, 13420, 12342, 15730, 14191, 17515, 17106, 19601

Offset: 0

Author

N. J. A. Sloane, Jan 27 2001

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000203, A000292, A001157, A001158, A002129, A007437, A013662, A048272, A073570, A101289.

Programs

Maple

a:= proc(n) option remember;
      add(d*(d+1)*(d+2)/6, d=numtheory[divisors](n))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 12 2023

Mathematica

With[{nn=50},CoefficientList[Series[Sum[x^n/(1-x^n)^4,{n,nn}],{x,0,nn}],x]] (* Harvey P. Dale, May 14 2013 *)

PARI

a(n) = if(n==0, 0, sumdiv(n, d, binomial(d+2, 3))); \\ Seiichi Manyama, Apr 19 2021

PARI

a(n) = if(n==0, 0, my(f = factor(n)); (sigma(f, 3) + 3*sigma(f, 2) + 2 * sigma(f)) / 6); \\ Amiram Eldar, Dec 29 2024

Formula

a(n) = (1/6)*(sigma_3(n) + 3*sigma_2(n) + 2*sigma_1(n)), i.e., this sequence is the inverse Möbius transform of tetrahedral (or pyramidal) numbers: n*(n+1)(n+2)/6 with g.f. 1/(1-x)^4 (cf. A000292). - Vladeta Jovovic, Aug 31 2002

L.g.f.: -log(Product_{k>=1} (1 - x^k)^((k+1)*(k+2)/6)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 21 2018

From Amiram Eldar, Dec 29 2024: (Start)

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

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