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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 (list; graph; refs; listen; history; text; internal format)
 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 Vincenzo Librandi, Table of n, a(n) for n = 0..300 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 Cf. A122045, A190339, A233808. Sequence in context: A056484 A056474 A219912 * A134912 A165836 A183827 Adjacent sequences:  A240977 A240978 A240979 * A240981 A240982 A240983 KEYWORD sign,frac AUTHOR Paul Curtz, Aug 06 2014 STATUS approved

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Last modified August 20 10:05 EDT 2019. Contains 326149 sequences. (Running on oeis4.)