A101289 Inverse Moebius transform of 5-simplex numbers A000389.

1, 7, 22, 63, 127, 280, 463, 855, 1309, 2135, 3004, 4704, 6189, 9037, 11776, 16359, 20350, 27901, 33650, 44695, 53614, 68790, 80731, 103776, 118882, 148701, 171220, 210469, 237337, 292292, 324633, 393351, 438922, 522298, 576346, 690333, 749399

Offset

1,2

Links

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

Formula

a(n) = Sum_{d|n} d*(d+1)*(d+2)*(d+3)*(d+4)/120 = Sum_{d|n} C(d+4,5) = Sum{d|n} A000389(d) = Sum_{d|n} (d^5+10*d^4+35*d^3+50*d^2+24*d)/120.

G.f.: Sum_{k>=1} x^k/(1 - x^k)^6 = Sum_{k>=1} binomial(k+4,5) * x^k/(1 - x^k). - Seiichi Manyama, Apr 19 2021

From Amiram Eldar, Dec 30 2024: (Start)

a(n) = (sigma_5(n) + 10*sigma_4(n) + 35*sigma_3(n) + 50*sigma_2(n) + 24*sigma_1(n)) / 120.

Dirichlet g.f.: zeta(s) * (zeta(s-5) + 10*zeta(s-4) + 35*zeta(s-3) + 50*zeta(s-2) + 24*zeta(s-1)) / 120.

Sum_{k=1..n} a(k) ~ (zeta(6)/720) * n^6. (End)

Prog

(PARI) a(n) = sumdiv(n, d, binomial(d+4, 5)); \\ Seiichi Manyama, Apr 19 2021
(PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+4, 5)*x^k/(1-x^k))) \\ Seiichi Manyama, Apr 19 2021
(PARI) a(n) = my(f = factor(n));  (sigma(f, 5) + 10*sigma(f, 4) + 35*sigma(f, 3) + 50*sigma(f, 2) + 24*sigma(f))/120; \\ Amiram Eldar, Dec 30 2024

Crossrefs

See also: A007437 (inverse Moebius transform of triangular numbers), A116963 (inverse Moebius transform of tetrahedral numbers).

Cf. A000389, A007437, A013664, A073570, A116963, A332508.

Cf. A000203, A001157, A001158, A001159, A001160.

Sequence in context: A246831 A354430 A122238 * A085287 A278767 A286186

Adjacent sequences: A101286 A101287 A101288 * A101290 A101291 A101292

Keyword

easy,nonn

Author

Jonathan Vos Post, Mar 31 2006

Status

approved