OFFSET
0,1
COMMENTS
1/(n+1) and Bernoulli(n,1) are autosequences in the sense that they remain the same (up to sign) under inverse binomial transform. This feature is kept for their sum, a(n)/A174342(n) = 2, 1, 1/2, 1/4, 1/6, 1/6, 1/6, 1/8, 7/90, 1/10, ...
Similar autosequences are also A000045, A001045, A113405, A000975 preceded by two zeros, and A140096.
Conjecture: the numerator of (A164555(n)/(n+1) + A027642(n)/(n+1)^2) is a(n) and the denominator of this fraction is equal to 1 if and only if n+1 is prime or 1. Cf. A309132. - Thomas Ordowski, Jul 09 2019
The "if" part of the conjecture is true: see the theorems in A309132 and A326690. The values of the numerator when n+1 is prime are A327033. - Jonathan Sondow, Aug 15 2019
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300
OEIS Wiki, Autosequence
MAPLE
B := proc(n) if n <> 1 then bernoulli(n) ; else -bernoulli(n) ; end if; end proc:
A174341 := proc(n) B(n)+1/(n+1) ; numer(%) ; end proc: # R. J. Mathar, Nov 19 2010
MATHEMATICA
a[n_] := Numerator[BernoulliB[n, 1] + 1/(n + 1)];
Table[a[n], {n, 0, 47}] (* Peter Luschny, Jul 13 2019 *)
PROG
(PARI)
B(n)=if(n!=1, bernfrac(n), -bernfrac(n));
a(n)=numerator(B(n) + 1/(n + 1));
for(n=0, 50, print1(a(n), ", ")) \\ Indranil Ghosh, Jun 19 2017
(Python)
from sympy import bernoulli, Integer
def B(n): return bernoulli(n) if n != 1 else -bernoulli(n)
def a(n): return (B(n) + 1/Integer(n + 1)).numerator() # Indranil Ghosh, Jun 19 2017
(Magma) [2, 1] cat [Numerator(Bernoulli(n)+1/(n+1)): n in [2..40]]; // Vincenzo Librandi, Jul 18 2019
CROSSREFS
KEYWORD
sign,frac
AUTHOR
Paul Curtz, Mar 16 2010
EXTENSIONS
Reformulation of the name by Peter Luschny, Jul 13 2019
STATUS
approved