A216705 a(n) = Product_{k=1..n} (81 - 9/k).

1, 72, 5508, 429624, 33832890, 2679564888, 213025408596, 16981168285224, 1356370816782267, 108509665342581360, 8691624193940766936, 696910230823250585232, 55927046023565859464868, 4491372003738673637024784, 360913821729000560118063000

Offset: 0

Author

Michel Lagneau, Sep 16 2012

Keywords

• nonn

Comments

This sequence is generalizable: Product_{k=1..n} (q^2 - q/k) = (q^n/n!) * Product_{k=0..n-1} (q*k + q-1) = expansion of (1- x*q^2)^((1-q)/q).

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..524

Crossrefs

Cf. A004988, A049382, A004994, A216702, A216703, A216704.

Programs

Maple

seq(product(81-9/k, k=1.. n), n=0..20);
seq((9^n/n!)*product(9*k+8, k=0.. n-1), n=0..20);

Mathematica

Table[Product[81-9/k,{k,n}],{n,0,20}] (* Harvey P. Dale, Jul 20 2021 *)

Formula

From Amiram Eldar, Aug 17 2025: (Start)

a(n) = 81^n * Gamma(n+8/9) / (Gamma(8/9) * Gamma(n+1)).

a(n) ~ c * 81^n / n^(1/9), where c = 1/Gamma(8/9) = 0.927851... . (End)