A370014 a(n) = Product_{k=0..n} (2^k + 4^(n-k)).

2, 15, 510, 84240, 69204960, 284844384000, 5892302096179200, 613826012249992396800, 322003239202740297793536000, 850857971372280730568060043264000, 11334246342025651164429104024534384640000, 760681528794595483313206024106936185273712640000

Offset

0,1

Comments

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.

Links

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

Formula

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

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

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

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

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

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

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

Example

a(0) = (1 + 1) = 2;
a(1) = (1 + 4)*(2 + 1) = 15;
a(2) = (1 + 4^2)*(2 + 4)*(2^2 + 1) = 510;
a(3) = (1 + 4^3)*(2 + 4^2)*(2^2 + 4)*(2^3 + 1) = 84240;
a(4) = (1 + 4^4)*(2 + 4^3)*(2^2 + 4^2)*(2^3 + 4)*(2^4 + 1) = 69204960;
a(5) = (1 + 4^5)*(2 + 4^4)*(2^2 + 4^3)*(2^3 + 4^2)*(2^4 + 4)*(2^5 + 1) = 284844384000;
...
RELATED SERIES.
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...

Prog

(PARI) {a(n) = prod(k=0, n, 2^k + 4^(n-k))}
for(n=0, 15, print1(a(n), ", "))

Crossrefs

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

Sequence in context: A013098 A365534 A158109 * A357856 A177394 A367770

Adjacent sequences: A370011 A370012 A370013 * A370015 A370016 A370017

Keyword

nonn

Author

Paul D. Hanna, Feb 08 2024

Status

approved