A238235 Numerators of Euler twin numbers Et(n).

1, -1, -1, -1, 1, 1, -1, -17, 17, 31, -31, -691, 691, 5461, -5461, -929569, 929569, 3202291, -3202291, -221930581, 221930581, 4722116521, -4722116521, -968383680827, 968383680827, 14717667114151, -14717667114151

Offset: 0

Author

Paul Curtz, Feb 20 2014

Keywords

• sign

Comments

Et(n) = 1, -1/2, -1/2, -1/4, 1/4, 1/2, -1/2, -17/8, 17/8, 31/2, -31/2, -691/4, 691/4, 5461/2, -5461/2,... =a(n)/b(n) is mentioned in A233808.

Denominators: b(n)= 1, 2, 2, 4, 4, 2, 2, 8, 8,... = A065176(n) with 1 instead of 0.

Et(n) is the first difference of 0, followed by A198631(n)/A006519(n+1).

Et(n+2) = -1/2, -1/4, 1/4, 1/2,... is an autosequence of the second kind. Its main diagonal is the double of the following diagonal, the inverse binomial transform of Et(n+2) being Et(n+2) signed.

The denominators of the difference table of Et(n+2) are even numbers of the form 2^p. For the Bernoulli twin numbers A051716(n+1)/A051717(n+2), the denominators of the difference table, A168426(n), are multiples of 3.

Crossrefs

Cf. A051716/A051717 (Bernoulli twin numbers).

Programs

Mathematica

Join[{1, -1, -1}, Table[{nu = Numerator[EulerE[2*n+1, 1]], -nu}, {n, 1, 12}]] // Flatten (* Jean-François Alcover, Feb 24 2014 *)

Formula

Binomial transform of A141424(n)/(A053644(n) with 1 instead of 0).

a(2n+3) = (-1)^n*A002425(n+2) = -a(2n+4).