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

%I A168516 #15 Aug 27 2022 03:14:56
%S A168516 -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,
%T A168516 -1,-1,-1,-8,-4,4,8,1,1,5,7,-4,-116,-32,-116,-4,7,5,5,5,32,28,16,-16,
%U A168516 -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
%N A168516 Table of the numerators of the fractions of Bernoulli twin numbers and their higher-order differences, read by antidiagonals.
%C A168516 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):
%C A168516 1, -1/2, -1/3, -1/6, -1/30, 1/30, 1/42, -1/42, -1/30, 1/30, 5/66, -5/66, ...
%C A168516 -3/2, 1/6, 1/6, 2/15, 1/15, -1/105, -1/21, -1/105, 1/15, 7/165, -5/33, ...
%C A168516 5/3, 0, -1/30, -1/15, -8/105, -4/105, 4/105, 8/105, -4/165, -32/165, ...
%C A168516 -5/3, -1/30, -1/30, -1/105, 4/105, 8/105, 4/105, -116/1155, -28/165, ...
%C A168516 49/30, 0, 1/42, 1/21, 4/105, -4/105, -32/231, -16/231, 5072/15015, 8128/15015, ...
%C A168516 -49/30, 1/42, 1/42, -1/105, -8/105, -116/1155, 16/231, 6112/15015, ...
%C A168516 Remove the two leftmost columns:
%C A168516 -1/3, -1/6, -1/30,  1/30, 1/42, -1/42, -1/30, 1/30, 5/66, -5/66,-691/2730, 691/2730, ...
%C A168516 1/6, 2/15, 1/15, -1/105, -1/21, -1/105, 1/15, 7/165, -5/33, -2663/15015, 691/1365, ...
%C A168516 -1/30, -1/15, -8/105, -4/105, 4/105, 8/105, -4/165, -32/165, -388/15015, 10264/15015, ...
%C A168516 -1/30, -1/105, 4/105, 8/105, 4/105, -116/1155, -28/165, 2524/15015, ...
%C A168516 1/42, 1/21, 4/105, -4/105, -32/231, -16/231, 5072/15015, 8128/15015, -2960/3003, ...
%C A168516 1/42, -1/105, -8/105, -116/1155, 16/231, 6112/15015, 3056/15015, -22928/15015, -7184/3003, ...
%C A168516 -1/30, -1/15, -4/165, 28/165, 5072/15015, -3056/15015, -3712/2145, ...
%C A168516 -1/30, 7/165, 32/165, 2524/15015, -8128/15015, -22928/15015, ...
%C A168516 and read the numerators upwards along antidiagonals to obtain the current sequence.
%C A168516 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.
%C A168516 In that sense, the C(n) with n>=2 are an eigensequence of the inverse binomial transform (i.e., an autosequence).
%p A168516 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:
%p A168516 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:
%p A168516 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
%t A168516 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 *)
%Y A168516 Cf. A168426 (denominators), A085737, A085738.
%K A168516 frac,tabl,sign
%O A168516 0,5
%A A168516 _Paul Curtz_, Nov 28 2009
%E A168516 Edited and extended by _R. J. Mathar_, Jul 10 2011