A290770 a(n) = Product_{k=1..n} k^(2*k).

1, 1, 16, 11664, 764411904, 7464960000000000, 16249593066946560000000000, 11020848942410302096869949440000000000, 3102093199396597590886754340698424229232640000000000, 465607547420733489126893933985879279492195953053596584509440000000000

Offset: 0

Author

Ilya Gutkovskiy, Aug 10 2017

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..27
• Eric Weisstein's World of Mathematics, Hyperfactorial
• Index entries for sequences related to factorial numbers

Crossrefs

Cf. A000178, A001044, A001818, A002109, A051675, A055209, A061742, A062206, A074962, A184877, A185141, A260122.

Programs

Magma

[1] cat [(&*[k^(2*k): k in [1..n]]): n in [1..10]]; // G. C. Greubel, Oct 14 2018

Mathematica

Table[Product[k^(2 k), {k, 1, n}], {n, 0, 9}]
Table[Hyperfactorial[n]^2, {n, 0, 9}]
Table[n!^(2 n)/BarnesG[n + 1]^2, {n, 0, 9}]

PARI

a(n) = prod(k=1, n, k^(2*k)) \\ Felix Fröhlich, Aug 10 2017

Formula

a(n) = A002109(n)^2.

a(n) = A185141(n)/A000178(n-1)^2 for n > 0.

a(n) = (n!)^(2*n)/G(n+1)^2, where G() is the Barnes G-function.

a(n) ~ A^2*exp(-n^2/2)*n^(n*(n+1))*n^(1/6), where A is the Glaisher-Kinkelin constant (A074962).