A368623 a(n) = Product_{k=1..n} (k^2 + 2*n^2).

1, 3, 108, 11286, 2337984, 804305700, 414285404544, 298436020283016, 286455044544970752, 353358684943164351792, 544692796454778554880000, 1025983872949208210500475232, 2318663822077115453077590638592, 6191980828123077577798830642106944, 19289639610614384872295428226588737536

Offset: 0

Author

Vaclav Kotesovec, Jan 01 2024

Keywords

• nonn

Comments

In general, for d>0, Product_{k=1..n} (k^2 + d*n^2) ~ (d+1)^(n + 1/2) * exp(n*(sqrt(d)*(Pi - 2*arctan(sqrt(d))) - 2)) * n^(2*n) / sqrt(d). - Vaclav Kotesovec, Jan 06 2024

Crossrefs

Cf. A272244, A324403, A324425.

Programs

Mathematica

Table[Product[k^2 + 2*n^2, {k, 1, n}], {n, 0, 20}]

Formula

a(n) ~ 3^(n + 1/2) * exp(n*(sqrt(2)*arctan(2*sqrt(2)) - 2)) * n^(2*n) / sqrt(2).