A369674 a(n) = Product_{k=0..n} (3^k + 3^(n-k)).

2, 16, 600, 112896, 108928800, 544431476736, 14105702277360000, 1900051576637594075136, 1328360485647389567734080000, 4830166933124609654538067824869376, 91168969237139220357818392868757600000000, 8950497893393998236587417126220897399198550327296

Offset

0,1

Links

Table of n, a(n) for n=0..11.

Formula

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

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

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

From Vaclav Kotesovec, Feb 07 2024: (Start)

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

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

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

Example

a(0) = (1 + 1) = 2;
a(1) = (1 + 3)*(3 + 1) = 16;
a(2) = (1 + 3^2)*(3 + 3)*(3^2 + 1) = 600;
a(3) = (1 + 3^3)*(3 + 3^2)*(3^2 + 3)*(3^3 + 1) = 112896;
a(4) = (1 + 3^4)*(3 + 3^3)*(3^2 + 3^2)*(3^3 + 3)*(3^4 + 1) = 108928800;
a(5) = (1 + 3^5)*(3 + 3^4)*(3^2 + 3^3)*(3^3 + 3^2)*(3^4 + 3)*(3^5 + 1) = 544431476736;
...
RELATED SERIES.
Let F(x) be the g.f. of A369557, then
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...

Prog

(PARI) {a(n) = prod(k=0, n, 3^k + 3^(n-k))}
for(n=0, 15, print1(a(n), ", "))

Crossrefs

Cf. A369673, A369675, A369676, A369557.

Sequence in context: A012726 A013177 A060279 * A012757 A012464 A277036

Adjacent sequences: A369671 A369672 A369673 * A369675 A369676 A369677

Keyword

nonn

Author

Paul D. Hanna, Feb 06 2024

Status

approved