cp's OEIS Frontend

A263394 a(n) = Product_{i=1..n} (3^i - 2^i).

%I A263394 #59 Sep 08 2022 08:46:14
%S A263394 1,5,95,6175,1302925,866445125,1784010512375,11248186280524375,
%T A263394 215638979183932793125,12512451767147700321078125,
%U A263394 2190917791975795178520458609375,1155369543009475708416871245360859375,1832567448623162714866960405275465241328125
%N A263394 a(n) = Product_{i=1..n} (3^i - 2^i).
%C A263394 Generally, for sequences of the form a(n) = Product_{i=1..n} j^i-k^i, where j>k>=1 and  n>=1: given probability p=(k/j)^n that an outcome will occur at the n-th stage of an infinite process, then r = 1 - a(n)/j^((n^2+n)/2) is the probability that the outcome has occurred up to and including the n-th iteration. Here, j=3 and k=2, so p=(2/3)^n and r = 1-a(n)/A047656(n+1). The limiting ratio of r ~ 0.9307279.
%F A263394 a(n) = Product_{i=1..n} A001047(i).
%F A263394 a(n) ~ c * 3^(n*(n+1)/2), where c = QPochhammer(2/3) = 0.0692720728018644... . - _Vaclav Kotesovec_, Oct 10 2016
%p A263394 A263394:=n->mul(3^i-2^i, i=1..n): seq(A263394(n), n=1..15); # _Wesley Ivan Hurt_, Mar 02 2016
%t A263394 Table[Product[3^i - 2^i, {i, n}], {n, 15}] (* _Wesley Ivan Hurt_, Mar 02 2016 *)
%t A263394 FoldList[Times,Table[3^i-2^i,{i,15}]] (* _Harvey P. Dale_, Feb 06 2017 *)
%o A263394 (Magma) [&*[ 3^k-2^k: k in [1..n] ]: n in [1..16]]; // _Vincenzo Librandi_, Mar 03 2016
%o A263394 (PARI) a(n) = prod(k=1, n, 3^k-2^k); \\ _Michel Marcus_, Mar 05 2016
%Y A263394 Cf. A001047, A047656.
%Y A263394 Cf. sequences of the form Product_{i=1..n}(j^i - 1): A005329 (j=2), A027871 (j=3), A027637 (j=4), A027872 (j=5), A027873 (j=6), A027875 (j=7),A027876 (j=8), A027877 (j=9), A027878 (j=10), A027879 (j=11), A027880 (j=12).
%Y A263394 Cf. sequences of the form Product_{i=1..n}(j^i - k^1), k>1: A269576 (j=4, k=3), A269661 (j=5, k=4).
%K A263394 nonn,easy
%O A263394 1,2
%A A263394 _Bob Selcoe_, Mar 02 2016