A067055 -id:A067055 - cp's OEIS frontend

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

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.

A321581 a(n) is the sum of first n natural numbers raised to the product of first n natural numbers.

Original entry on oeis.org

0, 1, 9, 46656, 1000000000000000000000000

Offset: 0

Ayush Kumar Jha, Nov 13 2018

The next term a(5) has 142 decimal digits.

			a(3) = (1+2+3)^3! = 6^6 = 46656.

Cf. A000142, A000217, A067055.

List([1..5], n->(n*(n+1)/2)^(Factorial(n))); # Stefano Spezia, Nov 21 2018

[(n*(n+1)/2)^factorial(n)$n=0..5]; # Muniru A Asiru, Dec 20 2018

a[n_] := (n*(n + 1)/2)^n!; Array[a, 5] (* Amiram Eldar, Nov 13 2018 *)

a(n)=(n*(n+1)/2)^(n!); \\ Andrew Howroyd, Nov 13 2018

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

a(n) = A000217(n)^A000142(n).

A372117 a(n) = Product_{k=0..n} binomial(n+k, k)^k.

Original entry on oeis.org

1, 2, 108, 3200000, 1158107343750000, 119025168578031262646195453952, 82864944710388642300699757862681018776776867840000, 9481019710293786574190900386319772308050021208649248212215823364196925440000000

Offset: 0

Vaclav Kotesovec, Apr 19 2024

nonn

Cf. A067055, A255269, A306789, A362288, A372116.

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

a(n) = Product_{k=0..n} binomial(n + k, n)^k.
a(n) = A372116(n) / (A255269(n) * A067055(n)).
a(n) ~ 2^(2*n^3/3 + 3*n^2/4 + n/6 + 1/24) * exp(n^3/12 + n^2/4 - n/24 + zeta(3)/(8*Pi^2) - 1/24) / (sqrt(A) * Pi^(n^2/4 + n/4) * n^(n^2/4 + n/4 + 1/24)), where A is the Glaisher-Kinkelin constant A074962.

cp's OEIS Frontend

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

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A321581 a(n) is the sum of first n natural numbers raised to the product of first n natural numbers.

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A372117 a(n) = Product_{k=0..n} binomial(n+k, k)^k.

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