A062031 Group odd numbers into (1), (3,5,7), (9,11,13,15,17), ...; a(n) = product of n-th group.

1, 105, 328185, 5568833025, 304513870485825, 40992233865440682825, 11492457771692770753505625, 5984524775454356180393209490625, 5325142910343897163530366857379506625, 7598549164899334249502031499667984969915625

Offset: 1

Author

Amarnath Murthy, Jun 02 2001

Keywords

• nonn

Examples

			a(2) = 3*5*7 = 105.
		

Links

• Harry J. Smith, Table of n, a(n) for n=1..100

Crossrefs

Cf. A062029, A062030, A062032.

Programs

Mathematica

Table[(Gamma[2*n^2 +1]*Gamma[(n-1)^2 +1])/(2^(2*n-1)*Gamma[n^2 +1]*Gamma[2*(n-1)^2 +1]), {n, 30}] (* G. C. Greubel, May 06 2022 *)

PARI

a(n) = { my(b=2*n^2 - 4*n + 3); prod(k=0, 2*n - 2, b + 2*k) } \\ Harry J. Smith, Jul 30 2009

SageMath

[(gamma(2*n^2 +1)*gamma((n-1)^2 +1))/(2^(2*n-1)*gamma(n^2 +1)*gamma(2*(n-1)^2 +1)) for n in (1..30)] # G. C. Greubel, May 06 2022

Formula

a(n) = Product_{k=0..2*n-2} (2*k + 2*n*(n-2) + 3). - Harry J. Smith, Jul 30 2009

a(n) = (Gamma(2*n^2 + 1)*Gamma((n-1)^2 + 1))/(2^(2*n-1)*Gamma(n^2 + 1)*Gamma(2*(n-1)^2 + 1)). - G. C. Greubel, May 06 2022

a(n) ~ exp(-2) * 2^(2*n-1) * n^(4*n-2). - Vaclav Kotesovec, Jun 09 2025

Extensions

More terms from Matthew Conroy, Jun 11 2001