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a(n) = Product_{k=0..n} (5^k + 5^(n-k)).
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%I #16 Feb 07 2024 07:49:49

%S 2,36,6760,14288400,331135220000,87265295649000000,

%T 252668462115852250000000,8322480168806663555062500000000,

%U 3012058207750727786980181328125000000000,12401474551899042876552569922821191406250000000000,561039675887726306551826113078284190093383789062500000000000

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

%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. (End)

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

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

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

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

%F a(n) ~ c * 5^(3*n^2/4 + n), where

%F c = 2.170417138549358... = QPochhammer(-1, 1/25)^2/2 if n is even and

%F c = 2.189351749288445... = 5^(1/4) * QPochhammer(-5, 1/25)^2 / 36 if n is odd. (End)

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

%e a(1) = (1 + 5)*(5 + 1) = 36;

%e a(2) = (1 + 5^2)*(5 + 5)*(5^2 + 1) = 6760;

%e a(3) = (1 + 5^3)*(5 + 5^2)*(5^2 + 5)*(5^3 + 1) = 14288400;

%e a(4) = (1 + 5^4)*(5 + 5^3)*(5^2 + 5^2)*(5^3 + 5)*(5^4 + 1) = 331135220000;

%e a(5) = (1 + 5^5)*(5 + 5^4)*(5^2 + 5^3)*(5^3 + 5^2)*(5^4 + 5)*(5^5 + 1) = 87265295649000000;

%e ...

%e RELATED SERIES.

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

%e F(1/5) = 2 + 36/5^2 + 6760/5^6 + 14288400/5^12 + 331135220000/5^20 + 87265295649000000/5^30 + ... + a(n)/5^(n*(n+1)) + ... = 3.934732308501055907377639201049737298238369356...

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

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

%Y Cf. A369673, A369674, A369675, A369557.

%K nonn

%O 0,1

%A _Paul D. Hanna_, Feb 06 2024