A165161 Numerator of the n-th term in the first differences of the binomial transform of the "original" Bernoulli numbers.

1, 2, 5, 29, 31, 43, 41, 29, 31, 71, 61, 2039, 3421, 13, -1, -3107, 4127, 44665, -43069, -174281, 174941, 854651, -854375, -236361361, 236366821, 8553109, -8553097, -23749460159, 23749461899, 8615841290327

Offset: 0

Author

Paul Curtz, Sep 06 2009

Keywords

• frac
• sign

Comments

The binomial transform of the "original" Bernoulli numbers is 1, 3/2, 13/6, ... as mentioned in A164558.

The first differences of that sequence are 3/2 - 1 = 1/2, 13/6 - 3/2 = 2/3, 5/6, 29/30, 31/30, ... and the numerators of these differences are listed here.

The bisection a(2n) reappears (up to signs) as A162173(n+1).

Crossrefs

Cf. A051717 (denominators), A164555, A027642.

Programs

Maple

read("transforms") :
A164555 := proc(n) if n <= 2 then 1; else numer(bernoulli(n)) ; end if; end proc:
A027642 := proc(n) denom(bernoulli(n)) ; end proc:
nmax := 40:
BINOMIAL([seq(A164555(n)/A027642(n), n=0..nmax)]) :
map(numer,DIFF(%)) ; # R. J. Mathar, Jul 07 2011

Formula

a(2n) + A000367(n) = A006954(n+1) = A051717(2n+1).

a(2n+1) + a(2n+2) = A051717(2n+2) + A051717(2n+3), n > 0.