A015001 q-factorial numbers for q=3.

1, 1, 4, 52, 2080, 251680, 91611520, 100131391360, 328430963660800, 3232089113385932800, 95424198983606279987200, 8452007576574959037306265600, 2245867453247498115393020895232000, 1790317944898228845164815929864036352000

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Comments

a(n) is the number of maximal chains in the lattice of subspaces of an n-dimensional vector space over GF(3). - Geoffrey Critzer, Sep 07 2022

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..50
• Index entries for sequences related to factorial numbers.

Crossrefs

Cf. A015002, A015004, A015005, A015006, A015007, A015008, A015009, A015011.

Column q=3 of A069777.

Cf. A000079, A003462, A047656, A053290, A100220.

Programs

Magma

[n le 1 select 1 else (3^n-1)*Self(n-1)/2: n in [1..15]]; // Vincenzo Librandi, Oct 22 2012

Mathematica

RecurrenceTable[{a[1]==1, a[n]==((3^n - 1) * a[n-1])/2}, a, {n,15}] (* Vincenzo Librandi, Oct 27 2012 *)
Table[QFactorial[n, 3], {n, 15}] (* Bruno Berselli, Aug 14 2013 *)

Formula

a(n) = Product_{k=1..n} (q^k - 1) / (q - 1).

a(0) = 1, a(n) = (3^n - 1)*a(n-1)/2. - Vincenzo Librandi, Oct 27 2012

a(n) = (Product_{i=0..n-1} (q^n-q^i))/((q-1)^n*q^binomial(n,2)) = A053290(n)/(A000079(n)*A047656(n)). - Geoffrey Critzer, Sep 07 2022

From Amiram Eldar, Jul 05 2025: (Start)

a(n) = Product_{k=1..n} A003462(k).

a(n) ~ c * 3^(n*(n+1)/2)/2^n, where c = A100220. (End)

Extensions

a(0)=1 prepended by Alois P. Heinz, Sep 08 2021