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

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

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