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

0,4

Comments

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.

Links

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

Formula

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

Maple

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

Mathematica

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 *)

Crossrefs

Sequence in context: A111064 A071117 A054221 * A253318 A277026 A309592

Adjacent sequences: A195237 A195238 A195239 * A195241 A195242 A195243

Keyword

sign,frac

Author

Paul Curtz, Sep 13 2011

Status

approved