A369676 a(n) = Product_{k=0..n} (5^k + 5^(n-k)).

2, 36, 6760, 14288400, 331135220000, 87265295649000000, 252668462115852250000000, 8322480168806663555062500000000, 3012058207750727786980181328125000000000, 12401474551899042876552569922821191406250000000000, 561039675887726306551826113078284190093383789062500000000000

Offset: 0

Author

Paul D. Hanna, Feb 06 2024

Keywords

• nonn

Comments

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. (End)

Examples

			a(0) = (1 + 1) = 2;
a(1) = (1 + 5)*(5 + 1) = 36;
a(2) = (1 + 5^2)*(5 + 5)*(5^2 + 1) = 6760;
a(3) = (1 + 5^3)*(5 + 5^2)*(5^2 + 5)*(5^3 + 1) = 14288400;
a(4) = (1 + 5^4)*(5 + 5^3)*(5^2 + 5^2)*(5^3 + 5)*(5^4 + 1) = 331135220000;
a(5) = (1 + 5^5)*(5 + 5^4)*(5^2 + 5^3)*(5^3 + 5^2)*(5^4 + 5)*(5^5 + 1) = 87265295649000000;
...
RELATED SERIES.
Let F(x) be the g.f. of A369557, then
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...
		

Crossrefs

Cf. A369673, A369674, A369675, A369557.

Programs

PARI

{a(n) = prod(k=0, n, 5^k + 5^(n-k))}
for(n=0, 15, print1(a(n), ", "))

Formula

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

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

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

From Vaclav Kotesovec, Feb 07 2024: (Start)

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

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

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