This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A165161 #8 Feb 25 2019 08:21:56
%S A165161 1,2,5,29,31,43,41,29,31,71,61,2039,3421,13,-1,-3107,4127,44665,
%T A165161 -43069,-174281,174941,854651,-854375,-236361361,236366821,8553109,
%U A165161 -8553097,-23749460159,23749461899,8615841290327
%N A165161 Numerator of the n-th term in the first differences of the binomial transform of the "original" Bernoulli numbers.
%C A165161 The binomial transform of the "original" Bernoulli numbers is 1, 3/2, 13/6, ... as mentioned in A164558.
%C A165161 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 A165161 The bisection a(2n) reappears (up to signs) as A162173(n+1).
%F A165161 a(2n) + A000367(n) = A006954(n+1) = A051717(2n+1).
%F A165161 a(2n+1) + a(2n+2) = A051717(2n+2) + A051717(2n+3), n > 0.
%p A165161 read("transforms") :
%p A165161 A164555 := proc(n) if n <= 2 then 1; else numer(bernoulli(n)) ; end if; end proc:
%p A165161 A027642 := proc(n) denom(bernoulli(n)) ; end proc:
%p A165161 nmax := 40:
%p A165161 BINOMIAL([seq(A164555(n)/A027642(n), n=0..nmax)]) :
%p A165161 map(numer,DIFF(%)) ; # _R. J. Mathar_, Jul 07 2011
%Y A165161 Cf. A051717 (denominators), A164555, A027642.
%K A165161 frac,sign
%O A165161 0,2
%A A165161 _Paul Curtz_, Sep 06 2009