A367571 a(n) = Product_{k=0..n} (7*k)! / k!^7.

1, 5040, 3432645216000, 626489905645044080640000000, 41646279370357699257014919153469440000000000000, 1200992054275801322636044235924808416678612164215512865177600000000000000

Offset: 0

Author

Vaclav Kotesovec, Nov 23 2023

Keywords

• nonn

Comments

In general, for m > 1, Product_{k=0..n} (m*k)! / k!^m ~ A^(m - 1/m) * exp(m*n/2 - m/12 + 1/(12*m) - n/2) * m^(m*n^2/2 + m*n/2 - 1/(12*m) + n/2) * n^(-m*n/2 - m/3 + 1/(12*m) + n/2 + 1/4) * (2*Pi)^(-m*n/2 - m/4 + n/2 + 1/4) / Product_{j=1..m-1} Gamma(j/m)^(j/m), where A is the Glaisher-Kinkelin constant A074962.

Crossrefs

Cf. A000178, A071549, A271947.

Cf. A007685, A367567, A367568, A367569, A367570.

Programs

Mathematica

Table[Product[(7*k)!/k!^7, {k, 0, n}], {n, 0, 10}]
Table[Product[Binomial[7*k,k] * 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(7*k,k) * binomial(6*k,k) * binomial(5*k,k) * binomial(4*k,k) * binomial(3*k,k) * binomial(2*k,k).

a(n) = A271947(n) / A000178(n)^7.

a(n) ~ A^(48/7) * 7^(7*n^2/2 + 4*n - 1/84) * exp(3*n - 4/7) / (Gamma(1/7)^(1/7) * Gamma(2/7)^(2/7) * Gamma(3/7)^(3/7) * Gamma(4/7)^(4/7) * Gamma(5/7)^(5/7) * Gamma(6/7)^(6/7) * n^(3*n + 29/14) * (2*Pi)^(3*n + 3/2)), where A is the Glaisher-Kinkelin constant A074962.