A168516 Table of the numerators of the fractions of Bernoulli twin numbers and their higher-order differences, read by antidiagonals.

-1, 1, -1, -1, 2, -1, -1, -1, 1, 1, 1, -1, -8, -1, 1, 1, 1, 4, -4, -1, -1, -1, -1, 4, 8, 4, -1, -1, -1, -1, -8, -4, 4, 8, 1, 1, 5, 7, -4, -116, -32, -116, -4, 7, 5, 5, 5, 32, 28, 16, -16, -28, -32, -5, -5, -691, -2663, -388, 2524, 5072, 6112, 5072, 2524, -388, -2663, -691, -691, -691, -10264, -10652, -8128, -3056, 3056, 8128, 10652, 10264, 691, 691, 7, 1247, 556, -4148, -2960, -22928

Offset

0,5

Comments

Consider the Bernoulli twin numbers C(n) = A051716(n)/A051717(n) in the top row and successive higher order differences in the other rows of an array T(0,k) = C(k), T(n,k) = T(n-1,k+1)-T(n-1,k):

1, -1/2, -1/3, -1/6, -1/30, 1/30, 1/42, -1/42, -1/30, 1/30, 5/66, -5/66, ...

-3/2, 1/6, 1/6, 2/15, 1/15, -1/105, -1/21, -1/105, 1/15, 7/165, -5/33, ...

5/3, 0, -1/30, -1/15, -8/105, -4/105, 4/105, 8/105, -4/165, -32/165, ...

-5/3, -1/30, -1/30, -1/105, 4/105, 8/105, 4/105, -116/1155, -28/165, ...

49/30, 0, 1/42, 1/21, 4/105, -4/105, -32/231, -16/231, 5072/15015, 8128/15015, ...

-49/30, 1/42, 1/42, -1/105, -8/105, -116/1155, 16/231, 6112/15015, ...

Remove the two leftmost columns:

-1/3, -1/6, -1/30, 1/30, 1/42, -1/42, -1/30, 1/30, 5/66, -5/66,-691/2730, 691/2730, ...

1/6, 2/15, 1/15, -1/105, -1/21, -1/105, 1/15, 7/165, -5/33, -2663/15015, 691/1365, ...

-1/30, -1/15, -8/105, -4/105, 4/105, 8/105, -4/165, -32/165, -388/15015, 10264/15015, ...

-1/30, -1/105, 4/105, 8/105, 4/105, -116/1155, -28/165, 2524/15015, ...

1/42, 1/21, 4/105, -4/105, -32/231, -16/231, 5072/15015, 8128/15015, -2960/3003, ...

1/42, -1/105, -8/105, -116/1155, 16/231, 6112/15015, 3056/15015, -22928/15015, -7184/3003, ...

-1/30, -1/15, -4/165, 28/165, 5072/15015, -3056/15015, -3712/2145, ...

-1/30, 7/165, 32/165, 2524/15015, -8128/15015, -22928/15015, ...

and read the numerators upwards along antidiagonals to obtain the current sequence.

The leftmost column (i.e., the inverse binomial transform of the top row) in this chopped variant equals the top row up to a sign pattern (-1)^n.

In that sense, the C(n) with n>=2 are an eigensequence of the inverse binomial transform (i.e., an autosequence).

Links

Table of n, a(n) for n=0..83.

Maple

C := proc(n) if n=0 then 1; elif n mod 2 = 0 then bernoulli(n)+bernoulli(n-1); else -bernoulli(n)-bernoulli(n-1); end if; end proc:
A168516 := proc(n, k) L := [seq(C(i), i=0..n+k+3)] ; for c from 1 to n do L := DIFF(L) ; end do; numer(op(k+3, L)) ; end proc:
for d from 0 to 15 do for k from 0 to d do printf("%a, ", A168516(d-k, k)) ; end do: end do: # R. J. Mathar, Jul 10 2011

Mathematica

max = 13; c[0] = 1; c[n_?EvenQ] := BernoulliB[n] + BernoulliB[n-1]; c[n_?OddQ] := -BernoulliB[n] - BernoulliB[n-1]; cc = Table[c[n], {n, 0, max+1}]; diff = Drop[#, 2]& /@ Table[ Differences[cc, n], {n, 0, max-1}]; Flatten[ Table[ diff[[n-k+1, k]], {n, 1, max}, {k, 1, n}]] // Numerator (* Jean-François Alcover, Aug 09 2012 *)

Crossrefs

Cf. A168426 (denominators), A085737, A085738.

Sequence in context: A255404 A078090 A174341 * A294335 A194321 A194852

Adjacent sequences: A168513 A168514 A168515 * A168517 A168518 A168519

Keyword

frac,tabl,sign

Author

Paul Curtz, Nov 28 2009

Extensions

Edited and extended by R. J. Mathar, Jul 10 2011

Status

approved