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

0,10

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.

Links

Table of n, a(n) for n=0..36.

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.

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 *)

Crossrefs

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

Sequence in context: A047760 A276908 A242246 * A050925 A086696 A259743

Adjacent sequences: A229976 A229977 A229978 * A229980 A229981 A229982

Keyword

sign,frac

Author

Paul Curtz, Oct 05 2013

Extensions

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

Status

approved