A176618 -id:A176618 - cp's OEIS frontend

A195240 Numerators of the second differences of the sequence of fractions (-1)^(n+1)*A176618(n)/A172031(n).

0, 1, 1, 7, 8, 11, 10, 7, 8, 19, 14, 337, 1028, 5, -2, -1681, 1936, 22133, -21734, -87223, 87388, 427291, -427222, -118181363, 118182728, 4276553, -4276550, -11874730297, 11874730732, 4307920641583

Offset: 0

Paul Curtz, Sep 13 2011

The array of (-1)^n*A176328(n)/A176591(n) and its first, second, etc. differences in subsequence rows starts as follows:
0,     1,     2,   19/6,   14/3, 199/30,    137/15, ... (-1)^n * A176328(n)/A176591(n),
1,     1,   7/6,    3/2,  59/30,    5/2,    127/42, ... see A176328,
0,   1/6,   1/3,   7/15,   8/15,  11/21,     10/21, ...
1/6, 1/6,  2/15,   1/15, -1/105,  -1/21,    -1/105, ... see A190339
0, -1/30, -1/15, -8/105, -4/105,  4/105, -116/1155, ...
The numerators in the 3rd row, 0, 1/6, 1/3, 7/15, 8/15, 11/21, 10/21, 7/15, 8/15, 19/33, 14/33, 337/1365, 1028/1365, 5/3, -2/3, -1681/255, 1936/255, ... define the current sequence.
The associated denominators are 1, 6 and followed by 3, 15, 15 etc as provided in A172087.
The second column of the array, 1, 1, 1/6, 1/6, -1/30, -1/30, ... contains doubled A000367(n)/A002445(n). These are related to A176150, A176144, and A176184.
In the first subdiagonal of the array we see 1, 1/6, 2/15, -8/150, 8/105, -32/321, 6112/15015, -3712/2145 , ... continued as given by A181130 and A181131.

Vincenzo Librandi, Table of n, a(n) for n = 0..300

read("transforms") ;
evb := [0, 1, 0, seq(bernoulli(n), n=2..30)] ;
ievb := BINOMIALi(evb) ;
[seq((-1)^n*op(n,ievb),n=1..nops(ievb))] ;
DIFF(%) ;
DIFF(%) ;
apply(numer,%) ; # R. J. Mathar, Sep 20 2011

evb = Join[{0, 1, 0}, Table[BernoulliB[n], {n, 2, 32}]]; ievb = Table[ Sum[Binomial[n, k]*evb[[k+1]], {k, 0, n}], {n, 0, Length[evb]-3}]; Differences[ievb, 2] // Numerator (* Jean-François Alcover, Sep 09 2013, after R. J. Mathar *)

a(2*n+1) + a(2*n+2) = A172087(2*n+2) = A172087(2*n+3), n >= 1.

A256003 a(n) = 0 followed by numerators of 2*A176327(n)/A176289(n).

Original entry on oeis.org

0, 2, 0, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 7, 0, -3617, 0, 43867, 0, -174611, 0, 854513, 0, -236364091, 0, 8553103, 0, -23749461029, 0, 8615841276005, 0, -7709321041217, 0, 2577687858367, 0, -26315271553053477373, 0

Offset: 0

Paul Curtz, May 06 2015

Offset 0 is chosen instead of -1. (The offset 0 corresponds to A176327(n), -1 to 0 followed by A176327(n).)
Denominators: b(n) = 1 followed by A141459(n).
Difference table of a(n)/b(n):
0,          2,      0,   1/3,     0, -1/15, 0, ...
2,         -2,    1/3,  -1/3, -1/15,  1/15, ...
-4,       7/3,   -2/3,  4/15,  2/15, ...
19/3,      -3,  14/15, -2/15, ...
-28/3,  59/15, -16/15, ...
199/15,    -5, ...
-274/15, ...
etc.
Without the first column, the antidiagonal sums are (-1)^n * A254667(n+1).
The Bernoulli numbers A027641(n)/A027642(n) or A164555(n)/A027642(n) come from A000027. 0 followed by the Bernoulli numbers comes from A001477. a(0)=0 is a choice.

Cf. A176327/A176289, A172086/A172087, A141459, A176618, A195240, A254667.

a(2n) = 0. a(2n+1) = A172086(2n), from the main pure Bernoulli twin numbers.

cp's OEIS Frontend

A195240 Numerators of the second differences of the sequence of fractions (-1)^(n+1)*A176618(n)/A172031(n).

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