cp's OEIS Frontend

A369678 a(n) = Product_{k=0..n} (3^k + 5^(n-k)).

%I A369678 #12 Feb 09 2024 04:44:13
%S A369678 2,24,2080,1382976,7148699648,287041728769536,90391546425391144960,
%T A369678 221202979125273147766738944,4237647337376998325597017538035712,
%U A369678 633933934421036224259931934460116571357184,738292285249623417870561091674252758908330993254400
%N A369678 a(n) = Product_{k=0..n} (3^k + 5^(n-k)).
%F A369678 a(n) = Product_{k=0..n} (3^k + 5^(n-k)).
%F A369678 a(n) = 15^(n*(n+1)/2) * Product_{k=0..n} (1/3^k + 1/5^(n-k)).
%F A369678 a(n) = 5^(n*(n+1)/2) * Product_{k=0..n} (1 + 3^n/15^k).
%F A369678 a(n) = 3^(n*(n+1)/2) * Product_{k=0..n} (1 + 5^n/15^k).
%F A369678 a(n) = 3^(-n*(n+1)/2) * Product_{k=0..n} (3^n + 15^k).
%F A369678 a(n) = 5^(-n*(n+1)/2) * Product_{k=0..n} (5^n + 15^k).
%F A369678 a(n) = 3^(n*(n+1)/2)*QPochhammer(-5^n, 1/15, n + 1). - _Stefano Spezia_, Feb 07 2024
%F A369678 Limit_{n->oo} a(n)^(1/n^2) = 3^(1/(2*(1 + log(5)/log(3)))) * sqrt(5) = 5^(1/(2*(1 + log(3)/log(5)))) * sqrt(3) = 2.794249622709633938040980858655052416325961... - _Vaclav Kotesovec_, Feb 07 2024
%F A369678 Equivalently, limit_{n->oo} a(n)^(1/n^2) = exp((1/2) * (log(3)^2 + log(3)*log(5) + log(5)^2) / log(15)). - _Paul D. Hanna_, Feb 08 2024
%e A369678 a(0) = (1 + 1) = 2;
%e A369678 a(1) = (1 + 5)*(3 + 1) = 24;
%e A369678 a(2) = (1 + 5^2)*(3 + 5)*(3^2 + 1) = 2080;
%e A369678 a(3) = (1 + 5^3)*(3 + 5^2)*(3^2 + 5)*(3^3 + 1) = 1382976;
%e A369678 a(4) = (1 + 5^4)*(3 + 5^3)*(3^2 + 5^2)*(3^3 + 5)*(3^4 + 1) = 7148699648;
%e A369678 a(5) = (1 + 5^5)*(3 + 5^4)*(3^2 + 5^3)*(3^3 + 5^2)*(3^4 + 5)*(3^5 + 1) = 287041728769536;
%e A369678 ...
%e A369678 RELATED SERIES.
%e A369678 Sum_{n>=0} Product_{k=0..n} (1/3^k + 1/5^(n-k)) = 2 + 24/15 + 2080/15^3 + 1382976/15^6 + 7148699648/15^10 + 287041728769536/15^15 + ... + a(n)/15^(n*(n+1)/2) + ... = 4.3507806549816093424129450104392682482776...
%o A369678 (PARI) {a(n) = prod(k=0, n, 3^k + 5^(n-k))}
%o A369678 for(n=0, 15, print1(a(n), ", "))
%Y A369678 Cf. A369673, A369674, A369675, A369676, A369677, A369679, A369680.
%K A369678 nonn
%O A369678 0,1
%A A369678 _Paul D. Hanna_, Feb 07 2024