A073570 G.f.: Sum_{n >= 1} x^n/(1-x^n)^5.

1, 6, 16, 41, 71, 147, 211, 371, 511, 791, 1002, 1547, 1821, 2596, 3146, 4247, 4846, 6627, 7316, 9681, 10852, 13657, 14951, 19427, 20546, 25577, 27916, 34096, 35961, 44912, 46377, 56607, 59922, 70896, 74096, 90278, 91391, 108591, 113766, 133421

Offset: 1

Author

Vladeta Jovovic, Aug 31 2002

Keywords

• nonn

Comments

Inverse Moebius transform of pentatope numbers. - Jonathan Vos Post, Mar 31 2006

Links

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

Crossrefs

Cf. A000005, A000332, A007437, A013663, A059358, A101289.

Cf. A000203, A001157, A001158, A001159.

Programs

Mathematica

Table[(DivisorSigma[4,n]+6*DivisorSigma[3,n]+11*DivisorSigma[2,n]+ 6*DivisorSigma[ 1,n])/24,{n,40}] (* Harvey P. Dale, Aug 08 2013 *)

PARI

a(n) = sumdiv(n, d, binomial(d+3, 4)); \\ Seiichi Manyama, Apr 19 2021

PARI

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+3, 4)*x^k/(1-x^k))) \\ Seiichi Manyama, Apr 19 2021

PARI

a(n) = my(f = factor(n)); (sigma(f, 4) + 6*sigma(f, 3) + 11*sigma(f, 2) + 6*sigma(f)) / 24; \\ Amiram Eldar, Dec 30 2024

Formula

a(n) = (1/24) * (sigma_4(n) + 6*sigma_3(n) + 11*sigma_2(n) + 6*sigma_1(n)).

a(n) = Sum_{d|n} (d+1)*(d+2)*(d+3)*(d+4)/24 = Sum_{d|n} C(d+3,4) = Sum_{d|n} A000332(d+3). - Jonathan Vos Post, Mar 31 2006. Corrected by Joshua Zucker, May 04 2007

From Amiram Eldar, Dec 30 2024: (Start)

Dirichlet g.f.: zeta(s) * (zeta(s-4) + 6*zeta(s-3) + 11*zeta(s-2) + 6*zeta(s-2)) / 24.

Sum_{k=1..n} a(k) ~ (zeta(5)/120) * n^5. (End)

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 31 2007