cp's OEIS Frontend

A369680 a(n) = Product_{k=0..n} (2^k + 3^(n-k)).

%I A369680 #22 Feb 09 2024 04:45:05
%S A369680 2,12,250,19404,5780918,6691848108,30261978906250,535757771934053916,
%T A369680 37171553237849766044342,10113067879819381109893992732,
%U A369680 10789224041146220828897229003906250,45150513047221188662211059385153001179564,741117672560101894851755994230829254062662140918
%N A369680 a(n) = Product_{k=0..n} (2^k + 3^(n-k)).
%C A369680 For p > 1, q > 1, limit_{n->oo} (Product_{k=0..n} (p^k + q^(n-k)) )^(1/n^2) = q^(1/2)*p^(1/(2*(1 + log(q)/log(p)))) = p^(1/2)*q^(1/(2*(1 + log(p)/log(q)))). - _Vaclav Kotesovec_, Feb 07 2024
%C A369680 Equivalently, limit_{n->oo} ( Product_{k=0..n} (p^k + q^(n-k)) )^(1/n^2) = exp((1/2) * (log(p)^2 + log(p)*log(q) + log(q)^2) / log(p*q)). - _Paul D. Hanna_, Feb 08 2024
%F A369680 a(n) = Product_{k=0..n} (2^k + 3^(n-k)).
%F A369680 a(n) = 6^(n*(n+1)/2) * Product_{k=0..n} (1/2^k + 1/3^(n-k)).
%F A369680 a(n) = 3^(n*(n+1)/2) * Product_{k=0..n} (1 + 2^n/6^k).
%F A369680 a(n) = 2^(n*(n+1)/2) * Product_{k=0..n} (1 + 3^n/6^k).
%F A369680 a(n) = 2^(-n*(n+1)/2) * Product_{k=0..n} (2^n + 6^k).
%F A369680 a(n) = 3^(-n*(n+1)/2) * Product_{k=0..n} (3^n + 6^k).
%F A369680 a(n) = 2^(n*(n+1)/2)*QPochhammer(-3^n, 1/6, n + 1). - _Stefano Spezia_, Feb 07 2024
%F A369680 Limit_{n->oo} a(n)^(1/n^2) = 2^(1/(2*(1 + log(3)/log(2)))) * sqrt(3) = 3^(1/(2*(1 + log(2)/log(3)))) * sqrt(2) = 1.9805589654474717155611061670180902111915926... - _Vaclav Kotesovec_, Feb 07 2024
%F A369680 Equivalently, limit_{n->oo} a(n)^(1/n^2) = exp((1/2) * (log(2)^2 + log(2)*log(3) + log(3)^2) / log(6)). - _Paul D. Hanna_, Feb 08 2024
%e A369680 a(0) = (1 + 1) = 2;
%e A369680 a(1) = (1 + 3)*(2 + 1) = 12;
%e A369680 a(2) = (1 + 3^2)*(2 + 3)*(2^2 + 1) = 250;
%e A369680 a(3) = (1 + 3^3)*(2 + 3^2)*(2^2 + 3)*(2^3 + 1) = 19404;
%e A369680 a(4) = (1 + 3^4)*(2 + 3^3)*(2^2 + 3^2)*(2^3 + 3)*(2^4 + 1) = 5780918;
%e A369680 a(5) = (1 + 3^5)*(2 + 3^4)*(2^2 + 3^3)*(2^3 + 3^2)*(2^4 + 3)*(2^5 + 1) = 6691848108;
%e A369680 ...
%e A369680 RELATED SERIES.
%e A369680 Sum_{n>=0} Product_{k=0..n} (1/2^k + 1/3^(n-k)) = 2 + 12/6 + 250/6^3 + 19404/6^6 + 5780918/6^10 + 6691848108/6^15 + ... + a(n)/6^(n*(n+1)/2) + ... = 5.6846111010137973166832330595516662115250385271...
%t A369680 Table[Product[2^k+3^(n-k),{k,0,n}],{n,0,12}] (* _James C. McMahon_, Feb 07 2024 *)
%o A369680 (PARI) {a(n) = prod(k=0, n, 2^k + 3^(n-k))}
%o A369680 for(n=0, 15, print1(a(n), ", "))
%Y A369680 Cf. A369673, A369674, A369675, A369676, A369677, A369678, A369679.
%Y A369680 Cf. A323715.
%K A369680 nonn
%O A369680 0,1
%A A369680 _Paul D. Hanna_, Feb 06 2024