A059358 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

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

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

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 *)

PROG

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

CROSSREFS

Cf. A000203, A000292, A002129, A007437, A048272, A073570, A101289.