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

%I A027872 #58 May 07 2023 01:25:14
%S A027872 1,4,96,11904,7428096,23205371904,362560730628096,
%T A027872 28324694519589371904,11064305472020078810628096,
%U A027872 21609960560733744406929189371904,211034749490954911990173458030810628096
%N A027872 a(n) = Product_{i=1..n} (5^i - 1).
%C A027872 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
%H A027872 G. C. Greubel, <a href="/A027872/b027872.txt">Table of n, a(n) for n = 0..50</a>
%F A027872 4^n|a(n) for n >= 1. - _G. C. Greubel_, Nov 21 2015
%F A027872 a(n) ~ c * 5^(n*(n+1)/2), where c = Product_{k>=1} (1-1/5^k) = A100222 . - _Vaclav Kotesovec_, Nov 21 2015
%F A027872 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
%F A027872 a(n) = Product_{i=1..n} A024049(i). - _Michel Marcus_, Dec 27 2015
%F A027872 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
%F A027872 Sum_{n>=0} (-1)^n/a(n) = A100222. - _Amiram Eldar_, May 07 2023
%p A027872 A027872 := proc(n)
%p A027872         mul( 5^i-1, i=1..n) ;
%p A027872 end proc: # _R. J. Mathar_, Mar 12 2013
%t A027872 Table[Product[(5^k-1),{k,1,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jul 17 2015 *)
%t A027872 Abs@QPochhammer[5, 5, Range[0, 10]] (* _Vladimir Reshetnikov_, Nov 20 2015 *)
%t A027872 Join[{1},FoldList[Times,5^Range[10]-1]] (* _Harvey P. Dale_, Dec 28 2021 *)
%o A027872 (PARI) a(n) = prod(i=1, n, 5^i-1); \\ _Michel Marcus_, Nov 21 2015
%o A027872 (Magma) [1] cat [&*[ 5^k-1: k in [1..n] ]: n in [1..11]]; // _Vincenzo Librandi_, Dec 24 2015
%Y A027872 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).
%Y A027872 Cf. A109345, A100222.
%K A027872 nonn
%O A027872 0,2
%A A027872 _N. J. A. Sloane_