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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
0, 1, 1, 1, 0, 0, 1, 1, -1, -1, 15, 15, -169, -169, 10753, 10753, -28713, -28713, 1586789, 1586789, -27542974, -13771487, 4694573547, 4694573547, -60230569205, -60230569205, 7328718272473, 7328718272473, -1043166080490099, -1043166080490099, 343459524172314625, 343459524172314625
OFFSET
0,11
COMMENTS
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).
f(n), companion to A198631(n)/A006519(n+1), is an autosequence of the first kind.
The difference table of f(n) is:
0, 1/2, 1/2, 1/4, 0, 0, ...
1/2, 0, -1/4, -1/4, 0, 1/4, ...
-1/2, -1/4, 0, 1/4, 1/4, -3/8, ...
1/4, 1/4, 1/4, 0, -5/8, -5/8, ...
etc.
The main diagonal is 0's=A000004. The first two upper diagonal are equal.
a(n) are the numerators of f(n).
f(n) is the first sequence of the family of alternated autosequences of the first and of the second kind
0, 1/2, 1/2, 1/4, 0, 0, ...
1, 1/2, 0, -1/4, 0, 1/2, ... = A198631(n)/A006519(n+1),
0, -1/2, -1/2, 1/4, 1, -1/2, ...
-1, -1/2, 1, 7/4, -2, -8, ...
etc.
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).
The first column is 0 followed by A122045(n).
For the numerators of the second column see A241209(n).
LINKS
EXAMPLE
2*f(1) = 0 + 1, f(1) = 1/2;
2*f(2) = 1/2 + 1/2, f(2) = 1/2;
2*f(3) = 1/2 + 0, f(3) = 1/4.
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
sign,frac
AUTHOR
Paul Curtz, Aug 06 2014
STATUS
approved