A241209 -id:A241209 - cp's OEIS frontend

A326481 a(n) = E2_{n}(0) with E2_{n} the polynomials defined in A326480.

1, -2, 2, 4, -16, -32, 272, 544, -7936, -15872, 353792, 707584, -22368256, -44736512, 1903757312, 3807514624, -209865342976, -419730685952, 29088885112832, 58177770225664, -4951498053124096, -9902996106248192, 1015423886506852352

Offset: 0

Peter Luschny, Jul 12 2019

sign

For comments see A326480.

Bisections (up to signs): A000182, A012509.
Cf. A326480, A155585, A326481, A326482, A326483, A326484.
Cf. A063376, A241209.

# The function E2(n) is defined in A326480.
seq(subs(x=0, E2(n)), n=0..22);

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

Original entry on oeis.org

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

Paul Curtz, Aug 06 2014

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).

			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.

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A122045, A190339, A233808.

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 *)

cp's OEIS Frontend

A326481 a(n) = E2_{n}(0) with E2_{n} the polynomials defined in A326480.

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