|
%I #13 Feb 09 2024 04:43:55
%S 2,18,910,275562,509528318,5782203860202,403066704971309470,
%T 172986911139059942455818,457494980583771669025834718462,
%U 7445459859979605380607238308201858858,746155118699551878624986638597659812003763550,461066589238234272286243169377378506495126815749310922
%N a(n) = Product_{k=0..n} (2^k + 5^(n-k)).
%F a(n) = Product_{k=0..n} (2^k + 5^(n-k)).
%F a(n) = 10^(n*(n+1)/2) * Product_{k=0..n} (1/2^k + 1/5^(n-k)).
%F a(n) = 5^(n*(n+1)/2) * Product_{k=0..n} (1 + 2^n/10^k).
%F a(n) = 2^(n*(n+1)/2) * Product_{k=0..n} (1 + 5^n/10^k).
%F a(n) = 2^(-n*(n+1)/2) * Product_{k=0..n} (2^n + 10^k).
%F a(n) = 5^(-n*(n+1)/2) * Product_{k=0..n} (5^n + 10^k).
%F a(n) = 2^(n*(n+1)/2)*QPochhammer(-5^n, 1/10, n + 1). - _Stefano Spezia_, Feb 07 2024
%F Limit_{n->oo} a(n)^(1/n^2) = 2^(1/(2*(1 + log(5)/log(2)))) * sqrt(5) = 5^(1/(2*(1 + log(2)/log(5)))) * sqrt(2) = 2.481958590195459039209137154563963236753327... - _Vaclav Kotesovec_, Feb 07 2024
%F Equivalently, limit_{n->oo} a(n)^(1/n^2) = exp((1/2) * (log(2)^2 + log(2)*log(5) + log(5)^2) / log(10)). - _Paul D. Hanna_, Feb 08 2024
%e a(0) = (1 + 1) = 2;
%e a(1) = (1 + 5)*(2 + 1) = 18;
%e a(2) = (1 + 5^2)*(2 + 5)*(2^2 + 1) = 910;
%e a(3) = (1 + 5^3)*(2 + 5^2)*(2^2 + 5)*(2^3 + 1) = 275562;
%e a(4) = (1 + 5^4)*(2 + 5^3)*(2^2 + 5^2)*(2^3 + 5)*(2^4 + 1) = 509528318;
%e a(5) = (1 + 5^5)*(2 + 5^4)*(2^2 + 5^3)*(2^3 + 5^2)*(2^4 + 5)*(2^5 + 1) = 5782203860202;
%e ...
%e RELATED SERIES.
%e Sum_{n>=0} Product_{k=0..n} (1/2^k + 1/5^(n-k)) = 2 + 18/10 + 910/10^3 + 275562/10^6 + 509528318/10^10 + 5782203860202/10^15 + ... + a(n)/10^(n*(n+1)/2) + ... = 5.0427178660718059961260933841217518099...
%o (PARI) {a(n) = prod(k=0, n, 2^k + 5^(n-k))}
%o for(n=0, 15, print1(a(n), ", "))
%Y Cf. A369673, A369674, A369675, A369676, A369678, A369679, A369680.
%K nonn
%O 0,1
%A _Paul D. Hanna_, Feb 07 2024
| 