A268506 -id:A268506 - cp's OEIS frontend

A294323 a(n) = Product_{k=0..n} (5*k + 1)!.

1, 720, 28740096000, 601322989968949248000000, 30722158107023001697205508762501120000000000, 12389984031943899068723274670059592852478855603111854080000000000000000

Offset: 0

Vaclav Kotesovec, Oct 28 2017

nonn

Cf. A168467, A268506, A294318, A294320, A294324, A294325, A294326.

Table[Product[(5*k + 1)!, {k, 0, n}] , {n, 0, 10}]

a(n) ~ 2^(n/2 + 7/10) * 5^(5*n^2/2 + 4*n + 4/3) * n^(5*n^2/2 + 4*n + 83/60) * Pi^(n/2 + 3/5) * Gamma(2/5)^(1/5) / (A^(1/5) * (1 + sqrt(5))^(1/10) * Gamma(1/5)^(2/5) * exp(15*n^2/4 + 4*n - 1/60)), where A is the Glaisher-Kinkelin constant A074962.

A268506(n) * A294323(n) * A294324(n) * A294325(n) * A294326(n) = A000178(5*n+4).

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

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Nov 23 2023

nonn

Cf. A000178, A008978, A268506.

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

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}]

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)).

A272096 a(n) = Product_{k=0..n} (k*n)!.

Original entry on oeis.org

1, 1, 48, 1567641600, 9698137182219213471744000000, 21488900044302744250061179567064173417691432878080000000000000000

Offset: 0

Vaclav Kotesovec, Apr 20 2016

The next term has 126 digits.

Cf. A000178, A098694, A268504, A268505, A268506, A271946.

Cf. A255322, A272093.

Table[Product[(k*n)!, {k, 0, n}], {n, 0, 6}]

a(n) ~ A^n * n^(1/4 + 13*n/12 + n^2 + n^3) * (2*Pi)^(1/4 + n/2) / exp(n*(2 + 2*n + 3*n^2)/4), where A = A074962 is the Glaisher-Kinkelin constant.

cp's OEIS Frontend

A294323 a(n) = Product_{k=0..n} (5*k + 1)!.

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

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A272096 a(n) = Product_{k=0..n} (k*n)!.

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