This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A101289 #22 Aug 08 2025 00:49:30
%S A101289 1,7,22,63,127,280,463,855,1309,2135,3004,4704,6189,9037,11776,16359,
%T A101289 20350,27901,33650,44695,53614,68790,80731,103776,118882,148701,
%U A101289 171220,210469,237337,292292,324633,393351,438922,522298,576346,690333,749399
%N A101289 Inverse Moebius transform of 5-simplex numbers A000389.
%C A101289 From _Georg Fischer_, Aug 06 2025: (Start)
%C A101289 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:
%C A101289 m Axxxxxx resulting sequence
%C A101289 ------------------------------
%C A101289 2 A000217 A007437
%C A101289 3 A000292 A116963
%C A101289 4 A000332 A073570
%C A101289 5 A000389 A101289 (this sequence)
%C A101289 The other formulas generalize correspondingly.
%C A101289 A116989 uses A000579 and m=6 within a modified formula.
%C A101289 (End)
%H A101289 Seiichi Manyama, <a href="/A101289/b101289.txt">Table of n, a(n) for n = 1..10000</a>
%F A101289 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.
%F A101289 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
%F A101289 From _Amiram Eldar_, Dec 30 2024: (Start)
%F A101289 a(n) = (sigma_5(n) + 10*sigma_4(n) + 35*sigma_3(n) + 50*sigma_2(n) + 24*sigma_1(n)) / 120.
%F A101289 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.
%F A101289 Sum_{k=1..n} a(k) ~ (zeta(6)/720) * n^6. (End)
%o A101289 (PARI) a(n) = sumdiv(n, d, binomial(d+4, 5)); \\ _Seiichi Manyama_, Apr 19 2021
%o A101289 (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
%o A101289 (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
%Y A101289 Cf. A000389, A007437, A013664, A073570, A116963, A116969, A332508.
%Y A101289 Cf. A000203, A001157, A001158, A001159, A001160.
%K A101289 easy,nonn
%O A101289 1,2
%A A101289 _Jonathan Vos Post_, Mar 31 2006