A371642 -id:A371642 - cp's OEIS frontend

A272241 a(n) = Product_{k=0..n} ((n^2 + k)! / (n^2 - k)!).

1, 2, 7200, 474211584000, 1981999450972492922880000, 1401219961854040654113268364083200000000000, 370389015130516478011776928922387124162707119541939129548800000000

Offset: 0

Vaclav Kotesovec, Apr 23 2016

The next term has 95 digits.

Cf. A103207, A272164, A272238, A371642.

Table[Product[(n^2+k)!/(n^2-k)!,{k,0,n}],{n,0,7}]
Table[BarnesG[n^2 - n + 1]*BarnesG[n^2 + n + 2]/(BarnesG[n^2 + 2]*BarnesG[n^2 + 1]), {n, 0, 6}]

a(n) = A272238(n) / A272164(n).

a(n) ~ exp(5/12) * n^(2*n*(n+1)).

A371644 a(n) = Product_{k=0..n} binomial(n^2 + k^2, n^2 - k^2).

Original entry on oeis.org

1, 1, 10, 57915, 8235313944000, 1077099640691257742845893750, 4629575796245443900868634734946423885068807034000

Offset: 0

Vaclav Kotesovec, Mar 31 2024

nonn

Cf. A371603, A371624, A371642, A371643, A371645.

Table[Product[Binomial[n^2+k^2, n^2-k^2], {k, 0, n}], {n, 0, 8}]

a(n) = A371642(n) / A371645(n).
a(n) = A371643(n) / (A371624(n) * A371645(n)).
a(n) ~ c * exp(Pi*n^3/3 + Pi*n/4 + n) / (2^(2*n^3/3 + 3*n/2) * Pi^(n/2) * A^(2*n) * n^(7*n/6 - 1/4)), where c = 0.761512... = 2^(1/4) * A255504 * (c from A371603) / (c from A371645) and A is the Glaisher-Kinkelin constant A074962.

cp's OEIS Frontend

A272241 a(n) = Product_{k=0..n} ((n^2 + k)! / (n^2 - k)!).

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