A193476 The denominators of the Bernoulli secant numbers at odd indices.

2, 56, 992, 16256, 261632, 4192256, 67100672, 1073709056, 17179738112, 274877382656, 628292059136, 70368735789056, 1125899873288192, 18014398375264256, 288230375614840832, 4611686016279904256, 73786976286248271872, 1180591620683051565056

Offset: 0

Author

Peter Luschny, Aug 17 2011

Keywords

• nonn
• frac

Comments

Denominator of the coefficient [x^(2n)] of sec(x)*(2*n+1)!/(4*16^n-2*4^n), that is, a(n) is the denominator of A000364(n)*(2*n+1)/(4*16^n-2*4^n). [Edited by Altug Alkan, Apr 22 2018]

Numerators are A160143. [Corrected by Peter Luschny, Mar 18 2021]

A193475(n) = 4*16^n-2*4^n is similar, but differs at n = 10, 31, 52, 73, 77, 94, ...

Links

• Peter Luschny, The lost Bernoulli numbers.

Crossrefs

Cf. A000364, A160143, A160144, A193475.

Programs

Maple

gf := (f,n) -> coeff(series(f(x),x,n+4),x,n):
A193476 := n -> denom(gf(sec,2*n)*(2*n+1)!/(4*16^n - 2*4^n)):
seq(A193476(n), n = 0..17); # Altug Alkan, Apr 23 2018

Mathematica

a[n_] := Sum[Sum[Binomial[k, m] (-1)^(n+k)/(2^(m-1)) Sum[Binomial[m, j]*(2j - m)^(2n), {j, 0, m/2}]*(-1)^(k-m), {m, 0, k}], {k, 1, 2n}] (2n+1)/ (4*16^n - 2*4^n) // Denominator; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jun 26 2019, after Vladimir Kruchinin in A000364 *)

PARI

a(n) = denominator(subst(bernpol(2*n+1), 'x, 1/4)*2^(2*n+1)/(2^(2*n+1)-1)); \\ Altug Alkan, Apr 22 2018 after Charles R Greathouse IV at A000364