A369681 - OEIS

%I #7 Feb 08 2024 07:29:39

%S 2,30,3978,4987710,58712437962,6601051349841150,

%T 7017151861981535193738,70966047508527496843460412990,

%U 6820716704126571481897874317127918922,6205644698427009393117687864650447521113942270,53916867047490616763228279441645027173409633988839675658

%N a(n) = Product_{k=0..n} (4^k + 5^(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).

%F a(n) = Product_{k=0..n} (4^k + 5^(n-k)).

%F a(n) = 20^(n*(n+1)/2) * Product_{k=0..n} (1/4^k + 1/5^(n-k)).

%F a(n) = 5^(n*(n+1)/2) * Product_{k=0..n} (1 + 4^n/20^k).

%F a(n) = 4^(n*(n+1)/2) * Product_{k=0..n} (1 + 5^n/20^k).

%F a(n) = 4^(-n*(n+1)/2) * Product_{k=0..n} (4^n + 20^k).

%F a(n) = 5^(-n*(n+1)/2) * Product_{k=0..n} (5^n + 20^k).

%F Limit_{n->oo} a(n)^(1/n^2) = exp((1/2) * (log(4)^2 + log(4)*log(5) + log(5)^2) / log(20)) = 3.0816872899745614612763875038173884057052077... [from a formula by _Vaclav Kotesovec_].

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

%e a(1) = (1 + 5)*(4 + 1) = 30;

%e a(2) = (1 + 5^2)*(4 + 5)*(4^2 + 1) = 3978;

%e a(3) = (1 + 5^3)*(4 + 5^2)*(4^2 + 5)*(4^3 + 1) = 4987710;

%e a(4) = (1 + 5^4)*(4 + 5^3)*(4^2 + 5^2)*(4^3 + 5)*(4^4 + 1) = 58712437962;

%e a(5) = (1 + 5^5)*(4 + 5^4)*(4^2 + 5^3)*(4^3 + 5^2)*(4^4 + 5)*(4^5 + 1) = 6601051349841150;

%e ...

%e RELATED SERIES.

%e Sum_{n>=0} Product_{k=0..n} (1/4^k + 1/5^(n-k)) = 2 + 30/20 + 3978/20^3 + 4987710/20^6 + 58712437962/20^10 + 6601051349841150/20^15 + ... + a(n)/20^(n*(n+1)/2) + ... = 4.0811214259450988699292249336017494522520...

%o (PARI) {a(n) = prod(k=0, n, 4^k + 5^(n-k))}

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

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

%K nonn

%O 0,1

%A _Paul D. Hanna_, Feb 07 2024