A193476 - OEIS

%I #44 Jan 25 2024 04:10:41

%S 2,56,992,16256,261632,4192256,67100672,1073709056,17179738112,

%T 274877382656,628292059136,70368735789056,1125899873288192,

%U 18014398375264256,288230375614840832,4611686016279904256,73786976286248271872,1180591620683051565056

%N The denominators of the Bernoulli secant numbers at odd indices.

%C 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]

%C Numerators are A160143. [Corrected by _Peter Luschny_, Mar 18 2021]

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

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/TheLostBernoulliNumbers">The lost Bernoulli numbers.</a>

%p gf := (f,n) -> coeff(series(f(x),x,n+4),x,n):

%p A193476 := n -> denom(gf(sec,2*n)*(2*n+1)!/(4*16^n - 2*4^n)):

%p seq(A193476(n), n = 0..17); # _Altug Alkan_, Apr 23 2018

%t 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 *)

%o (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

%Y Cf. A000364, A160143, A160144, A193475.

%K nonn,frac

%O 0,1

%A _Peter Luschny_, Aug 17 2011