A324440 - OEIS

%I #21 Dec 02 2023 06:02:59

%S 1,2,8520192,956147263254051187507200,

%T 790929096572487518050439299107158612099228070051840000,

%U 266108022587896795750359251172229660295854509829286134803404773931312693787460334360985600000000000

%N a(n) = Product_{i=1..n, j=1..n} (i^7 + j^7).

%C For m>1, Product_{j=1..n, k=1..n} (j^m + k^m) ~ c(m) * exp(n*(n+1)*s(m) - m*n*(n-2)/2) * n^(m*(n^2 - 1/4)), where s(m) = Sum_{j>=1} (-1)^(j+1)/(j*(1 + m*j)) and c(m) is a constant (dependent only on m). Equivalently, s(m) = log(2) - HurwitzLerchPhi(-1, 1, 1 + 1/m). - _Vaclav Kotesovec_, Dec 01 2023

%F Limit_{n->oo} (a(n)^(1/n^2))/n^7 = 2^(3/2) * (cos(3*Pi/14) / tan(Pi/7))^sin(3*Pi/14) / ((cos(Pi/14)*tan(3*Pi/14))^sin(Pi/14) * (sin(Pi/7)*tan(Pi/14))^cos(Pi/7)) * exp((Pi/sin(Pi/7) - 21)/2) = 0.0334234967249533921390751418772468470887965377...

%F From _Vaclav Kotesovec_, Dec 01 2023: (Start)

%F a(n) ~ c * exp(n*(n+1)*s - 7*n*(n-2)/2) * n^(7*(n^2 - 1/4)), where

%F s = Sum_{j>=1} (-1)^(j+1)/(j*(1 + 7*j)) = Pi/(2*sin(Pi/7)) + 3*log(2)/2 - 7 - cos(Pi/7) * log(2*sin(Pi/14)^2) - log(2*sin(3*Pi/14)^2) * sin(Pi/14) + log(cos(3*Pi/14)*cos(Pi/7) / sin(Pi/7)) * sin(3*Pi/14) = 0.10150386842315637912206687298894641634315636548242136512503... and

%F c = 0.068056503846689328929612652207251071282623125565150941566636264805878144...

%F Equivalently, s = log(2) - HurwitzLerchPhi(-1, 1, 1 + 1/7). (End)

%p a:= n-> mul(mul(i^7 + j^7, i=1..n), j=1..n):

%p seq(a(n), n=0..5);  # _Alois P. Heinz_, Nov 26 2023

%t Table[Product[i^7+j^7, {i, 1, n}, {j, 1, n}], {n, 1, 6}]

%o (Python)

%o from math import prod, factorial

%o def A324440(n): return prod(i**7+j**7 for i in range(1,n) for j in range(i+1,n+1))**2*factorial(n)**7<<n # _Chai Wah Wu_, Nov 26 2023

%Y Cf. A079478, A324403, A324426, A324437, A324438, A324439, A367834.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Feb 28 2019

%E a(0)=1 prepended by _Alois P. Heinz_, Nov 26 2023