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A242376 Numerators of b(n) = b(n-1)/2 + 1/(2*n), b(0)=0.

%I A242376 #37 Nov 14 2022 09:55:11
%S A242376 0,1,1,5,1,4,13,151,16,83,73,1433,647,15341,28211,10447,608,19345,
%T A242376 18181,651745,771079,731957,2786599,122289917,14614772,140001721,
%U A242376 134354573,774885169,745984697,41711914513,80530073893,4825521853483
%N A242376 Numerators of b(n) = b(n-1)/2 + 1/(2*n), b(0)=0.
%C A242376 See the denominators in A241519.
%C A242376 b(n) = 0, 1/2, 1/2, 5/12, 1/3, 4/15, 13/60, 151/840, 16/105, 83/630, 73/630, ...  (Ta0(n) in A241269) is an autosequence of the first kind.
%H A242376 Ralf Stephan, <a href="/A242376/b242376.txt">Table of n, a(n) for n = 0..999</a>
%F A242376 0 = b(n)*(+b(n+1) - 4*b(n+2) + 4*b(n+3)) + b(n+1)*(-2*b(n+1) + 9*b(n+2) - 10*b(n+3)) + b(n+2)*(-2*b(n+2) + 4*b(n+3)) if n>=0. - _Michael Somos_, May 26 2014
%F A242376 b(n) = -Re(Phi(2, 1, n + 1)). - _Eric W. Weisstein_, Dec 11 2017
%F A242376 G.f. for b(n): -log(1-x)/(2*(1-x/2)). - _Vladimir Kruchinin_, Nov 14 2022
%e A242376 0, 1/2, 1/2, 5/12, 1/3, 4/15, 13/60, 151/840, 16/105, 83/630, 73/630, ...
%t A242376 Table[-Re[LerchPhi[2, 1, n + 1]], {n, 0, 20}] // Numerator (* _Eric W. Weisstein_, Dec 11 2017 *)
%t A242376 -Re[LerchPhi[2, 1, Range[20]]] // Numerator (* _Eric W. Weisstein_, Dec 11 2017 *)
%t A242376 RecurrenceTable[{b[n] == b[n - 1]/2 + 1/(2 n), b[0] == 0}, b[n], {n, 20}] // Numerator (* _Eric W. Weisstein_, Dec 11 2017 *)
%o A242376 (Sage)
%o A242376 def a():
%o A242376     b = n = 0
%o A242376     while True:
%o A242376         yield numerator(b)
%o A242376         n = n + 1
%o A242376         b = (b/2 + 1/(2*n)) # _Ralf Stephan_, May 18 2014
%Y A242376 Cf. A241519 (denominators).
%K A242376 nonn,frac
%O A242376 0,4
%A A242376 _Paul Curtz_, May 12 2014
%E A242376 a(14)-a(25) from _Jean-François Alcover_, May 12 2014
%E A242376 Corrected a(22) and a(24), more terms from _Ralf Stephan_, May 18 2014