A369674 - OEIS

%I #21 Feb 07 2024 07:46:27

%S 2,16,600,112896,108928800,544431476736,14105702277360000,

%T 1900051576637594075136,1328360485647389567734080000,

%U 4830166933124609654538067824869376,91168969237139220357818392868757600000000,8950497893393998236587417126220897399198550327296

%N a(n) = Product_{k=0..n} (3^k + 3^(n-k)).

%F a(n) = Product_{k=0..n} (3^k + 3^(n-k)).

%F a(n) = 3^(n*(n+1)) * Product_{k=0..n} (1/3^k + 1/3^(n-k)).

%F a(n) = 3^(n*(n+1)/2) * Product_{k=0..n} (1 + 1/3^(n-2*k)).

%F From _Vaclav Kotesovec_, Feb 07 2024: (Start)

%F a(n) ~ c * 3^(3*n^2/4 + n), where

%F c = 2.538295806020848... = QPochhammer(-1, 1/9)^2/2 if n is even and

%F c = 2.539569717896307... = 3^(1/4) * QPochhammer(-3, 1/9)^2 / 16 if n is odd. (End)

%e a(0) = (1 + 1) = 2;

%e a(1) = (1 + 3)*(3 + 1) = 16;

%e a(2) = (1 + 3^2)*(3 + 3)*(3^2 + 1) = 600;

%e a(3) = (1 + 3^3)*(3 + 3^2)*(3^2 + 3)*(3^3 + 1) = 112896;

%e a(4) = (1 + 3^4)*(3 + 3^3)*(3^2 + 3^2)*(3^3 + 3)*(3^4 + 1) = 108928800;

%e a(5) = (1 + 3^5)*(3 + 3^4)*(3^2 + 3^3)*(3^3 + 3^2)*(3^4 + 3)*(3^5 + 1) = 544431476736;

%e ...

%e RELATED SERIES.

%e Let F(x) be the g.f. of A369557, then

%e F(1/3) = 2 + 16/3^2 + 600/3^6 + 112896/3^12 + 108928800/3^20 + 544431476736/3^30 + 14105702277360000/3^42 + ... + a(n)/3^(n*(n+1)) + ... = 4.847274134844057155467506697748724715389597193...

%o (PARI) {a(n) = prod(k=0, n, 3^k + 3^(n-k))}

%o for(n=0, 15, print1(a(n), ", "))

%Y Cf. A369673, A369675, A369676, A369557.

%K nonn

%O 0,1

%A _Paul D. Hanna_, Feb 06 2024