A165161 - OEIS

%I #8 Feb 25 2019 08:21:56

%S 1,2,5,29,31,43,41,29,31,71,61,2039,3421,13,-1,-3107,4127,44665,

%T -43069,-174281,174941,854651,-854375,-236361361,236366821,8553109,

%U -8553097,-23749460159,23749461899,8615841290327

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

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

%C 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.

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

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

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

%p read("transforms") :

%p A164555 := proc(n) if n <= 2 then 1; else numer(bernoulli(n)) ; end if; end proc:

%p A027642 := proc(n) denom(bernoulli(n)) ; end proc:

%p nmax := 40:

%p BINOMIAL([seq(A164555(n)/A027642(n), n=0..nmax)]) :

%p map(numer,DIFF(%)) ; # _R. J. Mathar_, Jul 07 2011

%Y Cf. A051717 (denominators), A164555, A027642.

%K frac,sign

%O 0,2

%A _Paul Curtz_, Sep 06 2009