A255932 a(n) is the denominator of Gamma(n+1/2)^2/(2*n*Pi), the value of an integral with sinh in the denominator.

8, 64, 128, 2048, 2048, 16384, 32768, 1048576, 524288, 4194304, 8388608, 134217728, 134217728, 1073741824, 2147483648, 137438953472, 34359738368, 274877906944, 549755813888, 8796093022208, 8796093022208, 70368744177664, 140737488355328

Offset: 1

Author

Jean-François Alcover, Mar 11 2015

Keywords

• frac
• nonn

Comments

Conjecture: a(n) <= 2^(3*n). - Vaclav Kotesovec, Mar 11 2015

Examples

			1/8, 9/64, 75/128, 11025/2048, 178605/2048, 36018675/16384, 2608781175/32768, ...
		

Links

• MathOverflow, An identity involving an infinite integral with a sinh in the denominator

Crossrefs

Cf. A255931 (numerators), A292608.

Programs

Maple

seq(2^A292608(n), n=1..23); # Peter Luschny, Sep 23 2017

Mathematica

a[n_] := Gamma[n+1/2]^2/(2*n*Pi) // Denominator; Array[a, 30]
Table[(2*n)!^2 / (n * 2^(4*n+1) * n!^2), {n, 1, 20}] // Denominator (* Vaclav Kotesovec, Mar 11 2015 *)
b[n_] := 2*n + 1 + IntegerExponent[n,2]; Table[2^b[n], {n,1,23}] (* Peter Luschny, Sep 23 2017 *)

Formula

The n-th fraction also equals the n-th coefficient in the expansion of 2F1(1/2,1/2; 1; x) * n!*(n-1)!/2.

a(n) = 2^(2*n + 1 + valuation(n, 2)) = 2^A292608(n). - Peter Luschny, Sep 23 2017