A027875 - OEIS

%I #42 May 07 2023 01:24:13

%S 1,6,288,98496,236390400,3972777062400,467389275837235200,

%T 384914699001548351078400,2218956256804125934296760320000,

%U 89542886518308517126993353029713920000

%N a(n) = Product_{i=1..n} (7^i - 1).

%H G. C. Greubel, <a href="/A027875/b027875.txt">Table of n, a(n) for n = 0..50</a>

%F 2*(10)^(2m)|a(n) where 4*m <= n <= 4*m+3, for m >= 1. - _G. C. Greubel_, Nov 20 2015

%F a(n) ~ c * 7^(n*(n+1)/2), where c = Product_{k>=1} (1-1/7^k) = A132035 = 0.836795407089037871026729798146136241352436435876... . - _Vaclav Kotesovec_, Nov 21 2015

%F a(n) = 7^(binomial(n+1,2))*(1/7;1/7)_{n}, where (a;q)_{n} is the q-Pochhammer symbol. - _G. C. Greubel_, Dec 24 2015

%F a(n) = Product_{i=1..n} A024075(i). - _Michel Marcus_, Dec 27 2015

%F G.f.: Sum_{n>=0} 7^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 7^k*x). - _Ilya Gutkovskiy_, May 22 2017

%F Sum_{n>=0} (-1)^n/a(n) = A132035. - _Amiram Eldar_, May 07 2023

%t Abs@QPochhammer[7, 7, Range[0, 10]] (* _Vladimir Reshetnikov_, Nov 20 2015 *)

%t Table[Product[7^k-1,{k,n}],{n,0,10}] (* _Harvey P. Dale_, Jul 28 2022 *)

%o (PARI) a(n) = prod(i=1, n, 7^i-1); \\ _Michel Marcus_, Nov 21 2015

%o (Magma) [1] cat [&*[ 7^k-1: k in [1..n] ]: n in [1..11]]; // _Vincenzo Librandi_, Dec 24 2015

%Y Cf. A005329 (q=2), A027871 (q=3), A027637 (q=4), A027872 (q=5), A027873 (q=6), A027876 (q=8), A027877 (q=9), A027878 (q=10), A027879 (q=11), A027880 (q=12).

%Y Cf. A132035.

%K nonn

%O 0,2

%A _N. J. A. Sloane_