A027872 a(n) = Product_{i=1..n} (5^i - 1).

1, 4, 96, 11904, 7428096, 23205371904, 362560730628096, 28324694519589371904, 11064305472020078810628096, 21609960560733744406929189371904, 211034749490954911990173458030810628096

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Given probability p = 1/5^n that an outcome will occur at the n-th stage of an infinite process, then starting at n=1, 1 - a(n)/A109345(n+1) is the probability that the outcome has occurred at or before the n-th iteration. The limiting ratio is 1-A100222 ~ 0.2396672. - Bob Selcoe, Mar 01 2016

Links

• G. C. Greubel, Table of n, a(n) for n = 0..50

Crossrefs

Cf. A005329 (q=2), A027871 (q=3), A027637 (q=4), A027873 (q=6), A027875 (q=7), A027876 (q=8), A027877 (q=9), A027878 (q=10), A027879 (q=11), A027880 (q=12).

Cf. A109345, A100222.

Programs

Magma

[1] cat [&*[ 5^k-1: k in [1..n] ]: n in [1..11]]; // Vincenzo Librandi, Dec 24 2015

Maple

A027872 := proc(n)
        mul( 5^i-1, i=1..n) ;
end proc: # R. J. Mathar, Mar 12 2013

Mathematica

Table[Product[(5^k-1),{k,1,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 17 2015 *)
Abs@QPochhammer[5, 5, Range[0, 10]] (* Vladimir Reshetnikov, Nov 20 2015 *)
Join[{1},FoldList[Times,5^Range[10]-1]] (* Harvey P. Dale, Dec 28 2021 *)

PARI

a(n) = prod(i=1, n, 5^i-1); \\ Michel Marcus, Nov 21 2015

Formula

4^n|a(n) for n >= 1. - G. C. Greubel, Nov 21 2015

a(n) ~ c * 5^(n*(n+1)/2), where c = Product_{k>=1} (1-1/5^k) = A100222 . - Vaclav Kotesovec, Nov 21 2015

a(n) = 5^(binomial(n+1,2))*(1/5; 1/5){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 23 2015

a(n) = Product_{i=1..n} A024049(i). - Michel Marcus, Dec 27 2015

G.f.: Sum_{n>=0} 5^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 5^k*x). - Ilya Gutkovskiy, May 22 2017

Sum_{n>=0} (-1)^n/a(n) = A100222. - Amiram Eldar, May 07 2023