cp's OEIS Frontend

The fractions are g(n)=0, 0, 1, 3/2, 3/2, 5/4, 5/4, 7/4, 7/4, -3/8, -3/8, 121/8, 121/8, -1261/8, -1261/8, 20583/8, 20583/8, -888403/16, -888403/16,... . The denominators are 1, 1, followed by A053644(n+1).

g(n+2) - g(n+1) = A198631(n)/A006519(n+1).

The corresponding fractions to g(n) are f(n) in A165142(n).

g(n) differences table:

0, 0, 1, 3/2, 3/2, 5/4,

0, 1, 1/2, 0, -1/4, 0,

1, -1/2, -1/2, -1/4, 1/4, 1/2, Euler twin numbers (new),

-3/2, 0, 1/4, 1/2, 1/4, -1,

3/2, 1/4, 1/4, -1/4, -5/4, -5/8,

-5/4, 0, -1/2, -1, 5/8, 13/2, etc.

Like A198631(n)/A006519(n+1),g(n) is an autosequence of the second kind.

If we proceed, here for Euler polynomials, like in A233565 for Bernoulli polynomials, we obtain

1) A133138(n)/A007395(n) (unreduced form) or

2) A233508(n)/A232628(n) (reduced form),the first array in A133135.

The Bernoulli's corresponding fractions to 1) are A193815(n)/(A003056(n) with 1 instead of 0).

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.