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A240980 Numerators of f(n) with 2*f(n+1) = f(n) + A198631(n)/A006519(n+1), f(0)=0.

%I A240980 #28 Jul 09 2017 04:20:54
%S A240980 0,1,1,1,0,0,1,1,-1,-1,15,15,-169,-169,10753,10753,-28713,-28713,
%T A240980 1586789,1586789,-27542974,-13771487,4694573547,4694573547,
%U A240980 -60230569205,-60230569205,7328718272473,7328718272473,-1043166080490099,-1043166080490099,343459524172314625,343459524172314625
%N A240980 Numerators of f(n) with 2*f(n+1) = f(n) + A198631(n)/A006519(n+1), f(0)=0.
%C A240980 An autosequence is a sequence which has its inverse binomial transform equal to the signed sequence. (Examples: 1) A000045(n) is of the first kind. 2) 1/(n+1) is of the second kind).
%C A240980 f(n), companion to A198631(n)/A006519(n+1), is an autosequence of the first kind.
%C A240980 The difference table of f(n) is:
%C A240980      0,  1/2,  1/2,  1/4,    0,    0, ...
%C A240980    1/2,    0, -1/4, -1/4,    0,  1/4, ...
%C A240980   -1/2, -1/4,    0,  1/4,  1/4, -3/8, ...
%C A240980    1/4,  1/4,  1/4,    0, -5/8, -5/8, ...
%C A240980 etc.
%C A240980 The main diagonal is 0's=A000004. The first two upper diagonal are equal.
%C A240980 a(n) are the numerators of f(n).
%C A240980 f(n) is the first sequence of the family of alternated autosequences of the first and of the second kind
%C A240980    0,  1/2,  1/2,  1/4,  0,    0, ...
%C A240980    1,  1/2,    0, -1/4,  0,  1/2, ... = A198631(n)/A006519(n+1),
%C A240980    0, -1/2, -1/2,  1/4,  1, -1/2, ...
%C A240980   -1, -1/2,    1,  7/4, -2,   -8, ...
%C A240980 etc.
%C A240980 Like A164555(n)/A027642(n), A198631(n)/A006519(n+1) is an autosequence which has its main diagonal equal to the first upper diagonal multiplied by 2. See A190339(n).
%C A240980 The first column is 0 followed by A122045(n).
%C A240980 For the numerators of the second column see A241209(n).
%H A240980 Vincenzo Librandi, <a href="/A240980/b240980.txt">Table of n, a(n) for n = 0..300</a>
%e A240980 2*f(1) = 0 + 1, f(1) = 1/2;
%e A240980 2*f(2) = 1/2 + 1/2, f(2) = 1/2;
%e A240980 2*f(3) = 1/2 + 0, f(3) = 1/4.
%t A240980 Clear[f]; f[0] = 0; f[1] = 1/2; f[n_] := f[n] = (1/2)*(EulerE[n-1, 1]/2^IntegerExponent[n-1, 2] + f[n-1]); Table[f[n] // Numerator, {n, 0, 31}] (* _Jean-François Alcover_, Aug 06 2014 *)
%Y A240980 Cf. A122045, A190339, A233808.
%K A240980 sign,frac
%O A240980 0,11
%A A240980 _Paul Curtz_, Aug 06 2014