OFFSET
0,1
COMMENTS
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
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
FORMULA
a(n) = Product_{k=0..n} (2^k + 3^(n-k)).
a(n) = 6^(n*(n+1)/2) * Product_{k=0..n} (1/2^k + 1/3^(n-k)).
a(n) = 3^(n*(n+1)/2) * Product_{k=0..n} (1 + 2^n/6^k).
a(n) = 2^(n*(n+1)/2) * Product_{k=0..n} (1 + 3^n/6^k).
a(n) = 2^(-n*(n+1)/2) * Product_{k=0..n} (2^n + 6^k).
a(n) = 3^(-n*(n+1)/2) * Product_{k=0..n} (3^n + 6^k).
a(n) = 2^(n*(n+1)/2)*QPochhammer(-3^n, 1/6, n + 1). - Stefano Spezia, Feb 07 2024
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
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
EXAMPLE
a(0) = (1 + 1) = 2;
a(1) = (1 + 3)*(2 + 1) = 12;
a(2) = (1 + 3^2)*(2 + 3)*(2^2 + 1) = 250;
a(3) = (1 + 3^3)*(2 + 3^2)*(2^2 + 3)*(2^3 + 1) = 19404;
a(4) = (1 + 3^4)*(2 + 3^3)*(2^2 + 3^2)*(2^3 + 3)*(2^4 + 1) = 5780918;
a(5) = (1 + 3^5)*(2 + 3^4)*(2^2 + 3^3)*(2^3 + 3^2)*(2^4 + 3)*(2^5 + 1) = 6691848108;
...
RELATED SERIES.
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...
MATHEMATICA
Table[Product[2^k+3^(n-k), {k, 0, n}], {n, 0, 12}] (* James C. McMahon, Feb 07 2024 *)
PROG
(PARI) {a(n) = prod(k=0, n, 2^k + 3^(n-k))}
for(n=0, 15, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 06 2024
STATUS
approved