A043303 - OEIS

A043303

Numerator of B(4n+2)/(2n+1) where B(m) are the Bernoulli numbers.

%I #27 Jan 03 2025 23:29:56

%S 1,1,1,1,43867,77683,657931,1723168255201,151628697551,

%T 154210205991661,1520097643918070802691,25932657025822267968607,

%U 19802288209643185928499101,29149963634884862421418123812691,2913228046513104891794716413587449,396793078518930920708162576045270521

%N Numerator of B(4n+2)/(2n+1) where B(m) are the Bernoulli numbers.

%C Note that numerator of B(2n)/n is odd so B(2n)/(2n), B(2n)/(4n), etc. have the same numerators. - _Michael Somos_, Feb 01 2004

%D Bruce Berndt, Ramanujan's Notebooks Part II, Springer-Verlag; see Infinite series, p. 262.

%H Seiichi Manyama, <a href="/A043303/b043303.txt">Table of n, a(n) for n = 0..156</a>

%F B(4*n+2)/(8*n+4) = Sum_{k>=1} k^(4*n+1)/(exp(2*Pi*k)-1).

%F a(n) = A001067(2n+1).

%p seq(numer(bernoulli(4*n+2)/(2*n+1)),n=0..30); # _Robert Israel_, Sep 18 2016

%t Table[BernoulliB[4n+2]/(2n+1),{n,0,20}]//Numerator (* _Harvey P. Dale_, Aug 13 2018 *)

%o (PARI) a(n)=if(n<0,0,numerator(bernfrac(4*n+2)/(2*n+1)))

%Y Cf. A001067, A043304.

%K easy,frac,nonn

%O 0,5

%A _Benoit Cloitre_, Apr 04 2002