A367570 a(n) = Product_{k=0..n} (6*k)! / k!^6.

1, 720, 5388768000, 739474163011584000000, 2400828978003787120431882240000000000, 213271990853093812884314351984207293234859212800000000000, 569474121824212834327144127568532894901251393782268174537457286512640000000000000

Offset: 0

Author

Vaclav Kotesovec, Nov 23 2023

Keywords

• nonn

Crossrefs

Cf. A000178, A008979, A271946.

Cf. A007685, A367567, A367568, A367569, A367571.

Programs

Mathematica

Table[Product[(6*k)!/k!^6, {k, 0, n}], {n, 0, 10}]
Table[Product[Binomial[6*k,k] * 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(6*k,k) * binomial(5*k,k) * binomial(4*k,k) * binomial(3*k,k) * binomial(2*k,k).

a(n) = A271946(n) / A000178(n)^6.

a(n) ~ A^(35/6) * Gamma(1/3)^(5/3) * 2^(3*n^2 + n - 215/72) * 3^(3*n^2 + 7*n/2 + 47/72) * exp(5*n/2 - 35/72) / (n^(5*n/2 + 125/72) * Pi^(5*n/2 + 10/3)), where A is the Glaisher-Kinkelin constant A074962.