A277354 a(n) = Product_{k=1..n} (4*k^2+1).

1, 5, 85, 3145, 204425, 20646925, 2993804125, 589779412625, 151573309044625, 49261325439503125, 19753791501240753125, 9580588878101765265625, 5527999782664718558265625, 3742455852864014463945828125, 2937827844498251354197475078125

Offset: 0

Author

Vaclav Kotesovec, Oct 10 2016

Keywords

• nonn

Comments

In general, for m>0, Product_{k=1..n} (m*k^2+1) is asymptotic to 2*m^(n+1/2) * n^(2*n+1) * sinh(Pi/sqrt(m)) / exp(2*n).

Crossrefs

Cf. A079484, A101686, A101928, A274306, A277352, A277353.

Programs

Mathematica

Table[Product[4*k^2+1, {k, 1, n}], {n, 0, 15}]
Round@Table[2^(2 n + 1) Abs[Gamma[1 + I/2 + n]]^2 Sinh[Pi/2]/Pi, {n, 0, 15}] (* Vladimir Reshetnikov, Oct 10 2016 *)

PARI

a(n) = prod(k=1, n, (4*k^2+1)); \\ Michel Marcus, Oct 11 2016

Formula

a(n) = (-1)^(n+1) * A101928(n+2).

a(n) ~ 2^(2*n+2) * n^(2*n+1) * sinh(Pi/2) / exp(2*n).

a(n) = 2^(2*n+1) * |Gamma(1 + i/2 + n)|^2 * sinh(Pi/2)/Pi. - Vladimir Reshetnikov, Oct 10 2016