A327497 a(n) = Numerator([x^n] (4*sinh(sqrt(x)/2)^2*cosh(sqrt(x)))/x).

1, 7, 31, 127, 73, 2047, 8191, 4681, 131071, 524287, 42799, 8388607, 33554431, 19173961, 536870911, 2147483647, 53353631, 1108378657, 137438953471, 78536544841, 2199023255551, 8796093022207, 162139963543, 140737488355327, 562949953421311, 321685687669321

Offset: 0

Author

Peter Luschny, Oct 05 2019

Keywords

• nonn
• frac

Examples

			r(n) = 1, 7/12, 31/360, 127/20160, 73/259200, 2047/239500800, 8191/43589145600, ...
		

Crossrefs

Denominators in A327986.

Cf. A002675/A002677.

Programs

Maple

gf := (4*sinh(sqrt(x)/2)^2*cosh(sqrt(x)))/x: ser := series(gf, x, 40):
seq(numer(coeff(ser, x, n)), n=0..25);
# Alternative:
a := s -> (2*s + 1)!/(2^(2*s + 1) - 1):
seq(denom(a(n)), n = 0..25); # Peter Luschny, Jul 18 2021

Mathematica

a[s_] := ((1 - 2^(-1 - 2 s)) Pi^(-2 - 2 s) Cos[Pi s] Zeta[2 + 2 s])/(4 (1 - 2^(2 + 2 s)) Zeta[-1 - 2 s]);
Array[a, 26, 0] // Numerator (* Peter Luschny, Jun 13 2020 *)

Formula

a(n) = numerator([x^n] (cosh(2*sqrt(x)) - 2*cosh(sqrt(x)) + 1)/x).

a(n) = numerator (1/8)*cos(Pi*n)*Zeta(2*n+2)*Pi^(-2*n-2)/(-1+2^(2*n+2))*(-2+4^(-n))/Zeta(-1-2*n). - Peter Luschny, Jun 13 2020

a(n) = denominator((2*n + 1)!/(2^(2*n + 1) - 1)). - Peter Luschny, Jul 18 2021