A334835 a(n) = denominator (2^(4*n-1) * (2^(4*n-2) - 1) * (Bernoulli(4*n-2) / (4*n-2)!) * ((2*n-2)! / Euler(2*n-2))^2 ).

1, 15, 7875, 11174163, 209844223875, 2475721174255329, 123460585419481594375, 5779795241720954566935675, 3729407645972755442722659595875, 485491404557154927712860942825333525, 193817991848984690019014855170410665878125, 56920344781273501874745734859262004352327035925

Offset: 1

Author

Dimitris Valianatos, May 16 2020

Keywords

• nonn
• frac

Comments

See A334912.

Links

• X. Gourdon and P. Sebah, Some Constants from Number theory

Crossrefs

Cf. A000040, A065091, A334912 (numerators).

Cf. A000364, A027641/A027642.

Programs

Mathematica

Denominator[Table[2^(4*s - 1) * (2^(4*s - 2) - 1) * BernoulliB[4*s - 2] * (2*s - 2)!^2 / (EulerE[2*s - 2]^2 * (4*s - 2)!), {s, 1, 15}]] (* or *) Denominator[Table[(1 - 1/2^(4*s - 2))*Zeta[4*s - 2]/DirichletBeta[2*s - 1]^2, {s, 1, 15}]] (* Vaclav Kotesovec, May 17 2020 *)

PARI

E(n) = subst(bernpol(2*n+1), 'x, 1/4)*4^(2*n+1)*(-1)^(n+1)/(2*n+1); \\ see A000364
a(n) = denominator((2^(4*n-1)*(2^(4*n-2)-1)*(bernfrac(4*n-2)/(4*n-2)!)*((2*n-2)!/ E(n-1))^2)); \\ Michel Marcus, May 17 2020

Formula

a(n) = denominator (Product_{p = A065091, m_p = (p mod 4) - 2} ((p^(2*n - 1) + 1) / (p^(2*n - 1) - 1))^m_p) = denominator (2^(4*n) - 4) * ((2*n - 2)! / EulerE(2*n - 2))^2 * (zeta(4*n - 2) / Pi^(4*n - 2)).

a(n) = denominator((1 - 1/2^(4*n-2)) * zeta(4*n-2) / DirichletBeta(2*n-1)^2). - Vaclav Kotesovec, May 17 2020

Extensions

More terms from Michel Marcus, May 17 2020