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%I #12 Feb 09 2024 04:44:40
%S 2,20,1190,449540,1094267450,17283181758500,1774660248586902950,
%T 1182579046508766038251700,5134581376819479940742838299450,
%U 144547890423248529154421336209389168500,26527720524980501045637796065988864058001683750,31574745853363739268697794406696294745967395732336262500
%N a(n) = Product_{k=0..n} (3^k + 4^(n-k)).
%F a(n) = Product_{k=0..n} (3^k + 4^(n-k)).
%F a(n) = 12^(n*(n+1)/2) * Product_{k=0..n} (1/3^k + 1/4^(n-k)).
%F a(n) = 4^(n*(n+1)/2) * Product_{k=0..n} (1 + 3^n/12^k).
%F a(n) = 3^(n*(n+1)/2) * Product_{k=0..n} (1 + 4^n/12^k).
%F a(n) = 3^(-n*(n+1)/2) * Product_{k=0..n} (3^n + 12^k).
%F a(n) = 4^(-n*(n+1)/2) * Product_{k=0..n} (4^n + 12^k).
%F a(n) = 3^(n*(n+1)/2)*QPochhammer(-4^n, 1/12, n + 1). - _Stefano Spezia_, Feb 07 2024
%F Limit_{n->oo} a(n)^(1/n^2) = 3^(1/(2*(1 + log(4)/log(3)))) * 2 = 2^(1/(1 + log(3)/log(4))) * sqrt(3) = 2.54977004574388327607102436919328599299374003... - _Vaclav Kotesovec_, Feb 07 2024
%F Equivalently, limit_{n->oo} a(n)^(1/n^2) = exp((1/2) * (log(3)^2 + log(3)*log(4) + log(4)^2) / log(12)). - _Paul D. Hanna_, Feb 08 2024
%e a(0) = (1 + 1) = 2;
%e a(1) = (1 + 4)*(3 + 1) = 20;
%e a(2) = (1 + 4^2)*(3 + 4)*(3^2 + 1) = 1190;
%e a(3) = (1 + 4^3)*(3 + 4^2)*(3^2 + 4)*(3^3 + 1) = 449540;
%e a(4) = (1 + 4^4)*(3 + 4^3)*(3^2 + 4^2)*(3^3 + 4)*(3^4 + 1) = 1094267450;
%e a(5) = (1 + 4^5)*(3 + 4^4)*(3^2 + 4^3)*(3^3 + 4^2)*(3^4 + 4)*(3^5 + 1) = 17283181758500;
%e ...
%e RELATED SERIES.
%e Sum_{n>=0} Product_{k=0..n} (1/3^k + 1/4^(n-k)) = 2 + 20/12 + 1190/12^3 + 449540/12^6 + 1094267450/12^10 + 17283181758500/12^15 + ... + a(n)/12^(n*(n+1)/2) + ... = 4.5247082137580440222914164418070212438323...
%o (PARI) {a(n) = prod(k=0, n, 3^k + 4^(n-k))}
%o for(n=0, 15, print1(a(n), ", "))
%Y Cf. A369673, A369674, A369675, A369676, A369677, A369678, A369680.
%K nonn
%O 0,1
%A _Paul D. Hanna_, Feb 07 2024
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