A370014 - OEIS

%I #7 Feb 08 2024 03:50:32

%S 2,15,510,84240,69204960,284844384000,5892302096179200,

%T 613826012249992396800,322003239202740297793536000,

%U 850857971372280730568060043264000,11334246342025651164429104024534384640000,760681528794595483313206024106936185273712640000

%N a(n) = Product_{k=0..n} (2^k + 4^(n-k)).

%C For p > 1, q > 1, limit_{n->oo} ( Product_{k=0..n} (p^k + q^(n-k)) )^(1/n^2) = exp((1/2) * (log(p)^2 + log(p)*log(q) + log(q)^2) / log(p*q)); formula due to _Vaclav Kotesovec_ (cf. A369680). For this sequence, p = 2 and q = 4.

%F a(n) = Product_{k=0..n} (2^k + 4^(n-k)).

%F a(n) = 8^(n*(n+1)/2) * Product_{k=0..n} (1/2^k + 1/4^(n-k)).

%F a(n) = 4^(n*(n+1)/2) * Product_{k=0..n} (1 + 2^n/8^k).

%F a(n) = 2^(n*(n+1)/2) * Product_{k=0..n} (1 + 4^n/8^k).

%F a(n) = 2^(-n*(n+1)/2) * Product_{k=0..n} (2^n + 8^k).

%F a(n) = 4^(-n*(n+1)/2) * Product_{k=0..n} (4^n + 8^k).

%F Limit_{n->oo} a(n)^(1/n^2) = 2^(7/6) = 2.244924096618745962867... [using the formula by _Vaclav Kotesovec_ given in the comments section].

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

%e a(1) = (1 + 4)*(2 + 1) = 15;

%e a(2) = (1 + 4^2)*(2 + 4)*(2^2 + 1) = 510;

%e a(3) = (1 + 4^3)*(2 + 4^2)*(2^2 + 4)*(2^3 + 1) = 84240;

%e a(4) = (1 + 4^4)*(2 + 4^3)*(2^2 + 4^2)*(2^3 + 4)*(2^4 + 1) = 69204960;

%e a(5) = (1 + 4^5)*(2 + 4^4)*(2^2 + 4^3)*(2^3 + 4^2)*(2^4 + 4)*(2^5 + 1) = 284844384000;

%e ...

%e RELATED SERIES.

%e Sum_{n>=0} Product_{k=0..n} (1/2^k + 1/4^(n-k)) = 2 + 15/8 + 510/8^3 + 84240/8^6 + 69204960/8^10 + 284844384000/8^15 + 5892302096179200/8^21 + ... + a(n)/8^(n*(n+1)/2) + ... = 5.2656633442570372661094196585300212123165...

%o (PARI) {a(n) = prod(k=0, n, 2^k + 4^(n-k))}

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

%Y Cf. A369673, A369674, A369675, A369676, A369677, A369678, A369679, A369680, A369681.

%K nonn

%O 0,1

%A _Paul D. Hanna_, Feb 08 2024