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

Author

Jonathan Vos Post, Mar 31 2006

Keywords

• easy
• nonn

Comments

From Georg Fischer, Aug 06 2025: (Start)

The general pattern is a(n) = Sum_{d|n} (Product_{k=0..m-1} d+k)/m! = Sum_{d|n} binomial(d+m-1, m) = Sum{d|n} Axxxxxx(d), with:

m Axxxxxx resulting sequence

------------------------------

2 A000217 A007437

3 A000292 A116963

4 A000332 A073570

5 A000389 A101289 (this sequence)

The other formulas generalize correspondingly.

A116989 uses A000579 and m=6 within a modified formula.

(End)

Links

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

Crossrefs

Cf. A000389, A007437, A013664, A073570, A116963, A116969, A332508.

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

Programs

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

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)