OFFSET
0,1
COMMENTS
Conjectures:
(C.1) a(n) is a square iff n is not divisible by 4.
(C.2) a(2*n+1) is not divisible by 5 for n >= 0.
(C.3) exponent of highest power of 5 dividing a(4*n) = 2*A127428(n).
(C.4) exponent of highest power of 5 dividing a(4*n+2) = 2*A127428(n+1).
From Vaclav Kotesovec, Feb 07 2024: (Start)
For q > 1, Product_{k=0..n} (q^k + q^(n-k)) ~ c * q^(3*n^2/4 + n), where
c = QPochhammer(-1, 1/q^2)^2/2 if n is even and
c = q^(1/4) * QPochhammer(-q, 1/q^2)^2 / (q + 1)^2 if n is odd.
c_even / c_odd = EllipticTheta[2, 0, 1/q] / EllipticTheta[3, 0, 1/q] = JacobiTheta2(0, 1/q) / JacobiTheta3(0, 1/q). (End)
FORMULA
a(n) = Product_{k=0..n} (2^k + 2^(n-k)).
a(n) = 2^(n*(n+1)) * Product_{k=0..n} (1/2^k + 1/2^(n-k)).
a(n) = 2^(n*(n+1)/2)*QPochhammer(-2^n, 1/4, 1 + n). - Stefano Spezia, Feb 06 2024
From Vaclav Kotesovec, Feb 07 2024: (Start)
a(n) ~ c * 2^(3*n^2/4 + n), where
c = 3.676982087353134... = QPochhammer(-1, 1/4)^2/2 if n is even and
c = 3.676991719144565... = 2^(1/4) * QPochhammer(-2, 1/4)^2 / 9 if n is odd.
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)
EXAMPLE
a(0) = (1 + 1) = 2;
a(1) = (1 + 2)*(2 + 1) = 9;
a(2) = (1 + 2^2)*(2 + 2)*(2^2 + 1) = 100;
a(3) = (1 + 2^3)*(2 + 2^2)*(2^2 + 2)*(2^3 + 1) = 2916;
a(4) = (1 + 2^4)*(2 + 2^3)*(2^2 + 2^2)*(2^3 + 2)*(2^4 + 1) = 231200;
a(5) = (1 + 2^5)*(2 + 2^4)*(2^2 + 2^3)*(2^3 + 2^2)*(2^4 + 2)*(2^5 + 1) = 50808384;
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;
...
RELATED SERIES.
Let F(x) be the g.f. of A369557, then
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...
PROG
(PARI) {a(n) = prod(k=0, n, 2^k + 2^(n-k))}
for(n=0, 15, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 06 2024
STATUS
approved