A272093 -id:A272093 - cp's OEIS frontend

A272094 a(n) = Product_{k=0..n} binomial(k^2,k).

1, 1, 6, 504, 917280, 48735086400, 94925811409228800, 8154182636726616909619200, 36091760791026276649159689107865600, 9415901310649088228943246038670339934863360000, 162992165498634702043940163611264755298214594247272038400000

Offset: 0

Vaclav Kotesovec, Apr 20 2016

Cf. A255322, A272093, A272095.

Cf. A000178, A098694, A268196, A262261.

Table[Product[Binomial[k^2, k], {k, 0, n}], {n, 0, 10}]

a(n) = A255322(n) / (A272168(n) * A000178(n)).

a(n) ~ c1/c2 * A * exp(-1/12 + n/2 + n^2/4) * n^(1/12 + n^2/2) / (2*Pi)^(n/2), where c1 = Product_{k>=1} (k^2)!/stirling(k^2) = 1.14426047263759216966268786..., c2 = Product_{k>=2} (k*(k-1))!/stirling(k*(k-1)) = 1.086533635964823338078329..., stirling(n) = sqrt(2*Pi*n) * n^n / exp(n) is the Stirling approximation of n!, and A = A074962 is the Glaisher-Kinkelin constant.

A272095 a(n) = Product_{k=0..n} binomial(n^2,k).

Original entry on oeis.org

1, 1, 24, 27216, 1956864000, 11593630125000000, 7004354761049263478784000, 515246658615545697034849051407876096, 5368556637668593177532650186945239827409750982656, 9038577429104951379916309583338181472480254559457860096000000000

Offset: 0

Vaclav Kotesovec, Apr 20 2016

Cf. A272093, A272094, A272241.

Cf. A000178, A098694, A268196, A262261.

Table[Product[Binomial[n^2, k], {k, 0, n}], {n, 0, 10}]
Table[((n^2)!)^(n+1) * BarnesG[n^2 - n + 1] / (BarnesG[n^2 + 2] * BarnesG[n+2]), {n, 0, 10}]

a(n) = ((n^2)!)^(n+1) / (A272164(n) * A000178(n)).

a(n) ~ A * exp(3*n^2/4 + 5*n/6 - 1/8) * n^(n^2/2 - 5/12) / (2*Pi)^((n+1)/2), where A = A074962 is the Glaisher-Kinkelin constant.

A362288 a(n) = Product_{k=0..n} binomial(n,k)^k.

Original entry on oeis.org

1, 1, 2, 27, 9216, 312500000, 4251528000000000, 95432797246104853383515625, 14719075154533285649961930052505436160000, 65577306173662530591576256095315195684570038194755952705536

Offset: 0

Vaclav Kotesovec, Apr 14 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..22

Cf. A001142, A067055, A167008, A255268, A272093.

Table[Product[Binomial[n, k]^k, {k, 0, n}], {n, 0, 10}]
Table[(n!)^(n*(n+1)/2) / BarnesG[n+2]^n, {n, 0, 10}]

a(n) = prod(k=0, n, binomial(n,k)^k); \\ Michel Marcus, Apr 14 2023

a(n) = Product_{k=0..n} n!^k / k!^n.
a(n) = A067055(n) / A255268(n).
a(n) ~ A^n * exp((6*n^3 + 12*n^2 - n - 1)/24) / ((2*Pi)^(n*(n+1)/4) * n^(n*(3*n+2)/12)), where A is the Glaisher-Kinkelin constant A074962.

A345466 a(n) = Product_{k=1..n} binomial(n, floor(n/k)).

Original entry on oeis.org

1, 1, 2, 9, 96, 1250, 64800, 1764735, 224788480, 22499086176, 6123600000000, 408514437465750, 1308805762115174400, 133962125607455951520, 99335199198879310098432, 113040832521732593994140625, 425230288403106927476736000000, 72623663171934137824096600064000

Offset: 0

Vaclav Kotesovec, Jun 20 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 0..160

Cf. A010786, A034841, A051054, A092143, A272093.

[n eq 0 select 1 else (&*[Binomial(n,Floor(n/j)): j in [1..n]]): n in [0..30]]; // G. C. Greubel, Feb 05 2024

Table[Product[Binomial[n, Floor[n/k]], {k, 1, n}], {n, 0, 20}]
Table[Product[((n + 1)/k - 1)^Floor[n/k], {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 24 2021 *)

[product(binomial(n,(n//j)) for j in range(1,n+1)) for n in range(31)] # G. C. Greubel, Feb 05 2024

log(a(n)) ~ n * log(n)^2 / 2. - Vaclav Kotesovec, Jun 21 2021

a(n) = Product_{k=1..n} ((n+1)/k - 1)^floor(n/k). - Vaclav Kotesovec, Jun 24 2021

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

A272094 a(n) = Product_{k=0..n} binomial(k^2,k).

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A272095 a(n) = Product_{k=0..n} binomial(n^2,k).

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A362288 a(n) = Product_{k=0..n} binomial(n,k)^k.

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A345466 a(n) = Product_{k=1..n} binomial(n, floor(n/k)).

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

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