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%I #55 Sep 08 2022 08:45:51
%S 2,1,1,1,1,1,1,1,7,1,1,1,-37,1,37,1,-211,1,2311,1,-407389,1,37153,1,
%T -1181819909,1,76977929,1,-818946931,1,277930363757,1,-84802531453217,
%U 1,90219075042851,1,-711223555487930419,1,12696640293313423,1,-6367871182840222481,1,35351107998094669831,1,-83499808737903072705023,1,12690449182849194963361,1
%N a(n) = numerator(Bernoulli(n, 1) + 1/(n+1)).
%C a(n) is numerator of (A164555(n)/A027642(n) + 1/(n+1)).
%C 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, ...
%C Similar autosequences are also A000045, A001045, A113405, A000975 preceded by two zeros, and A140096.
%C 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
%C 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
%H Vincenzo Librandi, <a href="/A174341/b174341.txt">Table of n, a(n) for n = 0..300</a>
%H OEIS Wiki, <a href="https://oeis.org/wiki/Autosequence">Autosequence</a>
%p B := proc(n) if n <> 1 then bernoulli(n) ; else -bernoulli(n) ; end if; end proc:
%p A174341 := proc(n) B(n)+1/(n+1) ; numer(%) ;end proc: # _R. J. Mathar_, Nov 19 2010
%t a[n_] := Numerator[BernoulliB[n, 1] + 1/(n + 1)];
%t Table[a[n], {n, 0, 47}] (* _Peter Luschny_, Jul 13 2019 *)
%o (PARI)
%o B(n)=if(n!=1, bernfrac(n), -bernfrac(n));
%o a(n)=numerator(B(n) + 1/(n + 1));
%o for(n=0, 50, print1(a(n),", ")) \\ _Indranil Ghosh_, Jun 19 2017
%o (Python)
%o from sympy import bernoulli, Integer
%o def B(n): return bernoulli(n) if n != 1 else -bernoulli(n)
%o def a(n): return (B(n) + 1/Integer(n + 1)).numerator() # _Indranil Ghosh_, Jun 19 2017
%o (Magma) [2,1] cat [Numerator(Bernoulli(n)+1/(n+1)): n in [2..40]]; // _Vincenzo Librandi_, Jul 18 2019
%Y Cf. A164555, A027642, A174342 (denominators), A025529, A003506, A309132, A326690, A327033.
%K sign,frac
%O 0,1
%A _Paul Curtz_, Mar 16 2010
%E Reformulation of the name by _Peter Luschny_, Jul 13 2019
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