A369673 - OEIS

%I #21 Feb 08 2024 07:38:13

%S 2,9,100,2916,231200,50808384,31258240000,54112148361216,

%T 264265663201280000,3645603832850650497024,

%U 142153785549232537600000000,15673043740102659990892604030976,4886752115388739132874502963200000000,4309225323078788454199311474023086952546304,10747393363422494556085100202291563069440000000000

%N a(n) = Product_{k=0..n} (2^k + 2^(n-k)).

%C Conjectures:

%C (C.1) a(n) is a square iff n is not divisible by 4.

%C (C.2) a(2*n+1) is not divisible by 5 for n >= 0.

%C (C.3) exponent of highest power of 5 dividing a(4*n) = 2*A127428(n).

%C (C.4) exponent of highest power of 5 dividing a(4*n+2) = 2*A127428(n+1).

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

%C For q > 1, Product_{k=0..n} (q^k + q^(n-k)) ~ c * q^(3*n^2/4 + n), where

%C c = QPochhammer(-1, 1/q^2)^2/2 if n is even and

%C c = q^(1/4) * QPochhammer(-q, 1/q^2)^2 / (q + 1)^2 if n is odd.

%C c_even / c_odd = EllipticTheta[2, 0, 1/q] / EllipticTheta[3, 0, 1/q] = JacobiTheta2(0, 1/q) / JacobiTheta3(0, 1/q). (End)

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

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

%F a(n) = 2^(n*(n+1)/2)*QPochhammer(-2^n, 1/4, 1 + n). - _Stefano Spezia_, Feb 06 2024

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

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

%F c = 3.676982087353134... = QPochhammer(-1, 1/4)^2/2 if n is even and

%F c = 3.676991719144565... = 2^(1/4) * QPochhammer(-2, 1/4)^2 / 9 if n is odd.

%F c_even / c_odd = EllipticTheta[2, 0, 1/2] / EllipticTheta[3, 0, 1/2] = JacobiTheta2(0, 1/2) / JacobiTheta3(0, 1/2) = 0.9999973805240351337720926619... (End)

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

%e a(1) = (1 + 2)*(2 + 1) = 9;

%e a(2) = (1 + 2^2)*(2 + 2)*(2^2 + 1) = 100;

%e a(3) = (1 + 2^3)*(2 + 2^2)*(2^2 + 2)*(2^3 + 1) = 2916;

%e a(4) = (1 + 2^4)*(2 + 2^3)*(2^2 + 2^2)*(2^3 + 2)*(2^4 + 1) = 231200;

%e a(5) = (1 + 2^5)*(2 + 2^4)*(2^2 + 2^3)*(2^3 + 2^2)*(2^4 + 2)*(2^5 + 1) = 50808384;

%e a(6) = (1 + 2^6)*(2 + 2^5)*(2^2 + 2^4)*(2^3 + 2^3)*(2^4 + 2^2)*(2^5 + 2)*(2^6 + 1) = 31258240000;

%e ...

%e RELATED SERIES.

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

%e F(1/2) = 2 + 9/2^2 + 100/2^6 + 2916/2^12 + 231200/2^20 + 50808384/2^30 + 31258240000/2^42 + 54112148361216/2^56 + ... + a(n)/2^(n*(n+1)) + ... = 6.800139835051923542641455169580774467247971025...

%o (PARI) {a(n) = prod(k=0,n, 2^k + 2^(n-k))}

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

%Y Cf. A369674, A369675, A369676, A369557, A127428.

%K nonn

%O 0,1

%A _Paul D. Hanna_, Feb 06 2024