A162173 -id:A162173 - cp's OEIS frontend

A165226 Numerator of 1 - A164555(n)/A027642(n).

0, 1, 5, 1, 31, 1, 41, 1, 31, 1, 61, 1, 3421, 1, -1, 1, 4127, 1, -43069, 1, 174941, 1, -854375, 1, 236366821, 1, -8553097, 1, 23749461899, 1, -8615841261683, 1, 7709321041727, 1, -2577687858361, 1, 26315271553055396563, 1, -2929993913841553, 1

Offset: 0

Paul Curtz, Sep 09 2009

If n != 1, also the numerator of 1 - Bernoulli(n). The denominators are in A027642.
(There are no common factors to be canceled in the fractions.)
The numerators of 1 - Bernoulli(n) start 0, 3, 5,1, 31, ... and differ only at n=1 from this sequence.
E.g.f. for the rationals r(n) = a(n)/A027642(n) = 1 - A164555(n)/A027642(n): exp(x)*(1 - x/(exp(x) - 1)). - Wolfdieter Lang, Aug 07 2017

			The rationals r(n) begin: 0, 1/2, 5/6, 1, 31/30, 1, 41/42, 1, 31/30, 1, 61/66, 1, 3421/2730, 1, -1/6, 1, 4127/510, 1, -43069/798, 1, ... - _Wolfdieter Lang_, Aug 07 2017

Cf. A162173, A164555, A027642.

A165226 := proc(n) if n = 1 then 1+bernoulli(n) ; else 1-bernoulli(n) ; end if; numer(%) ; end proc: # R. J. Mathar, Jan 16 2011

|a(2n)| = A162173(n+1).

a(2n+1) = 1.

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

Original entry on oeis.org

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

Paul Curtz, Sep 06 2009

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).

Cf. A051717 (denominators), A164555, A027642.

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

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

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

cp's OEIS Frontend

A165226 Numerator of 1 - A164555(n)/A027642(n).

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