A367569 a(n) = Product_{k=0..n} (5*k)! / k!^5.

1, 120, 13608000, 2288430144000000, 699207483978843840000000000, 435858496811697532778806061260800000000000, 597507154003470929939550139366865942134606725120000000000000, 1898554530971015145216561379837863419725314413457243266261094236160000000000000000

Offset: 0

Author

Vaclav Kotesovec, Nov 23 2023

Keywords

• nonn

Crossrefs

Cf. A000178, A008978, A268506.

Cf. A007685, A367567, A367568, A367570, A367571.

Programs

Mathematica

Table[Product[(5*k)!/k!^5, {k, 0, n}], {n, 0, 10}]
Table[Product[Binomial[5*k,k] * Binomial[4*k,k] * Binomial[3*k,k] * Binomial[2*k,k], {k, 0, n}], {n, 0, 10}]

Formula

a(n) = Product_{k=0..n} binomial(5*k,k) * binomial(4*k,k) * binomial(3*k,k) * binomial(2*k,k).

a(n) = A268506(n) / A000178(n)^5.

a(n) ~ A^(24/5) * Gamma(1/5)^(3/5) * Gamma(2/5)^(2/5) * Gamma(3/5)^(1/5) * 5^(5*n^2/2 + 3*n + 23/60) * exp(2*n - 2/5) / (n^(2*n + 7/5) * (2*Pi)^(2*n + 13/5)), where A is the Glaisher-Kinkelin constant A074962.

Equivalently, a(n) ~ A^(24/5) * Gamma(1/5)^(3/5) * Gamma(2/5)^(1/5) * 5^(5*n^2/2 + 3*n + 1/3) * exp(2*n - 2/5) / ((1 + sqrt(5))^(1/10) * 2^(2*n + 23/10) * Pi^(2*n + 12/5) * n^(2*n + 7/5)).