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

%I #28 Jul 09 2017 04:20:54

%S 0,1,1,1,0,0,1,1,-1,-1,15,15,-169,-169,10753,10753,-28713,-28713,

%T 1586789,1586789,-27542974,-13771487,4694573547,4694573547,

%U -60230569205,-60230569205,7328718272473,7328718272473,-1043166080490099,-1043166080490099,343459524172314625,343459524172314625

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

%C 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 f(n), companion to A198631(n)/A006519(n+1), is an autosequence of the first kind.

%C The difference table of f(n) is:

%C 0, 1/2, 1/2, 1/4, 0, 0, ...

%C 1/2, 0, -1/4, -1/4, 0, 1/4, ...

%C -1/2, -1/4, 0, 1/4, 1/4, -3/8, ...

%C 1/4, 1/4, 1/4, 0, -5/8, -5/8, ...

%C etc.

%C The main diagonal is 0's=A000004. The first two upper diagonal are equal.

%C a(n) are the numerators of f(n).

%C f(n) is the first sequence of the family of alternated autosequences of the first and of the second kind

%C 0, 1/2, 1/2, 1/4, 0, 0, ...

%C 1, 1/2, 0, -1/4, 0, 1/2, ... = A198631(n)/A006519(n+1),

%C 0, -1/2, -1/2, 1/4, 1, -1/2, ...

%C -1, -1/2, 1, 7/4, -2, -8, ...

%C etc.

%C 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 The first column is 0 followed by A122045(n).

%C For the numerators of the second column see A241209(n).

%H Vincenzo Librandi, <a href="/A240980/b240980.txt">Table of n, a(n) for n = 0..300</a>

%e 2*f(1) = 0 + 1, f(1) = 1/2;

%e 2*f(2) = 1/2 + 1/2, f(2) = 1/2;

%e 2*f(3) = 1/2 + 0, f(3) = 1/4.

%t 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 Cf. A122045, A190339, A233808.

%K sign,frac

%O 0,11

%A _Paul Curtz_, Aug 06 2014

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)