A043303 Numerator of B(4n+2)/(2n+1) where B(m) are the Bernoulli numbers.

1, 1, 1, 1, 43867, 77683, 657931, 1723168255201, 151628697551, 154210205991661, 1520097643918070802691, 25932657025822267968607, 19802288209643185928499101, 29149963634884862421418123812691, 2913228046513104891794716413587449, 396793078518930920708162576045270521

Offset: 0

Author

Benoit Cloitre, Apr 04 2002

Keywords

• easy
• frac
• nonn

Comments

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

References

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..156

Crossrefs

Cf. A001067, A043304.

Programs

Maple

seq(numer(bernoulli(4*n+2)/(2*n+1)),n=0..30); # Robert Israel, Sep 18 2016

Mathematica

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

PARI

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

Formula

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

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