A277347 a(n) = Product_{k=1..n} (2*k*(k-1)+1).

1, 5, 65, 1625, 66625, 4064125, 345450625, 39035920625, 5660208490625, 1024497736803125, 226413999833490625, 59999709955875015625, 18779909216188879890625, 6854666863908941160078125, 2885814749705664228392890625, 1388076894608424493856980390625, 756501907561591349152054312890625

Offset: 1

Author

Michel Marcus, Oct 10 2016

Keywords

• nonn

Comments

Fang (see link) proves that a(n) is never a square for n > 1.

Links

• Jin-Hui Fang, Neither Product{k=1..n} (4k^2+1) nor Product{k=1..n} (2k(k-1)+1) is a perfect square, Integers, A16, Volume 9 (2009).

Crossrefs

Cf. A001844.

Programs

Mathematica

Table[Product[(2*k*(k-1)+1), {k,1,n}], {n, 1, 20}] (* Vaclav Kotesovec, Oct 10 2016 *)
Round@Table[2^n Abs[Gamma[1/2 + I/2 + n]]^2 Cosh[Pi/2]/Pi, {n, 1, 20}] (* Vladimir Reshetnikov, Oct 11 2016 *)
Rest@(CoefficientList[Series[HypergeometricPFQ[{1/2 - I/2, 1/2 + I/2}, {}, 2 x], {x, 0, 20}], x]*Range[0, 20]!) (* Benedict W. J. Irwin, Oct 19 2016 *)

PARI

a(n) = prod(k=1, n, 2*k*(k-1)+1);

Formula

a(n) ~ cosh(Pi/2) * 2^(n+1) * n^(2*n) / exp(2*n). - Vaclav Kotesovec, Oct 10 2016

a(n) = 2^n * |Gamma(1/2 + i/2 + n)|^2 * cosh(Pi/2)/Pi. - Vladimir Reshetnikov, Oct 11 2016

E.g.f.: 2F0((1-i)/2,(1+i)/2; ; 2*x). - Benedict W. J. Irwin, Oct 19 2016

a(n) = 2^(-1 + n)*Pochhammer(3/2 - i/2, -1 + n)*Pochhammer(3/2 + i/2, -1 + n), for n>=1. - Antonio Graciá Llorente, Sep 10 2023