A229979 Numerators of interleaved A063524(n) and A002427(n)/A006955(n).

0, 1, 1, 1, 0, -1, 0, 1, 0, -3, 0, 5, 0, -691, 0, 35, 0, -3617, 0, 43867, 0, -1222277, 0, 854513, 0, -1181820455, 0, 76977927, 0, -23749461029, 0, 8615841276005, 0, -84802531453387, 0, 90219075042845, 0

Offset: 0

Author

Paul Curtz, Oct 05 2013

Keywords

• sign
• frac

Comments

Numerators of Br(n) = 0, 1, 1, 1/2, 0, -1/6, 0, 1/6, 0, -3/10, 0, 5/6, 0, -691/210,... complementary Bernoulli numbers.

A164555(n)/A027642(n) is an autosequence of second kind. Its inverse binomial transform is the signed sequence and its main diagonal is the double of the first upper diagonal.

Br(n) is an autosequence of first kind. Its inverse binomial transform is the signed sequence and its main diagonal is A000004=0's.

Br(n) difference table:

0, 1, 1, 1/2, 0, -1/6,...

1, 0, -1/2, -1/2, -1/6, 1/6,... =A140351(n)/A140219(n)

-1, -1/2, 0, 1/3, 1/3, 0,...

1/2, 1/2, 1/3, 0, -1/3, -1/3,...

0, -1/6, -1/3, -1/3, 0, 8/15,...

-1/6, -1/6, 0, 1/3, 8/15, 0,... etc.

Crossrefs

Cf. A050925: a similar sequence, because 2*(n+1)*B(n) and (n+1)*B(n) have the same numerator.

Programs

Mathematica

a[0] = 0; a[1] = a[2] = 1; a[n_] := 2*n*BernoulliB[n-1] // Numerator; Table[a[n], {n, 0, 36}] (* Jean-François Alcover, Nov 25 2013 *)

Formula

a(2n)=A063524(n). a(2n+1)=A002427(n).

a(n) = numerators of n * b(n) with b(n)=0 followed by A164555(n)/A027642(n) = 0, 1, 1/2, 1/6, 0,... in A165142(n).

a(n+1) = numerators of Br(n+1) = Br(n) + A140351(n)/A140219(n), a(0)=Br(0)=0.

Extensions

Cross-ref. to A050925 by Jean-François Alcover, Dec 09 2013