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

Author

Paul Curtz, Nov 28 2009

Keywords

• frac
• tabl
• sign

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).

Crossrefs

Cf. A168426 (denominators), A085737, A085738.

Programs

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 *)

Extensions

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