A228954 Bisection of A195240(n).

1, 7, 11, 7, 19, 337, 5, -1681, 22133, -87223, 427291, -118181363, 4276553, -11874730297, 4307920641583, -3854660520481, 1288843929185, -13157635776526258889, 1464996956920781, -130541359248224557643

Offset: 0

Author

Paul Curtz, Sep 09 2013

Keywords

• sign

Comments

The first bisection is b(n) = 0, 1, 8, 10, 8, 14, 1028, -2, 1936, -21734,... .

a(n) and b(n) are twice linked to Bernoulli numbers (A027641(n+4) or A164555(n+4))/A027642(n+4).

Programs

Mathematica

evb = Join[{0, 1, 0}, Table[BernoulliB[n], {n, 2, 42}]]; ievb = Table[ Sum[Binomial[n, k]*evb[[k + 1]], {k, 0, n}], {n, 0, Length[evb] - 3}]; A195240 = Differences[ievb, 2] // Numerator; Partition[A195240, 2][[All, 2]]
(* or *)
A000367[n_] := BernoulliB[2*n] // Numerator; A001897[n_] := -2*(2^(2*n - 1) - 1)*BernoulliB[2*n] // Denominator; a[0] = 1; a[n_] := (A000367[n + 1] + A001897[n + 1])/2; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Sep 09 2013, after R. J. Mathar *)

Formula

A195240(2n+1).

a(n+1) = b(n+2) + A000367(n+2).

a(n+1) = A001897(n+2) - b(n+2).

2*a(n+1) = A000367(n+2) + A001897(n+2).

Extensions

More terms from Jean-François Alcover, Sep 09 2013