A168426 -id:A168426 - cp's OEIS frontend

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

Paul Curtz, Nov 28 2009

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).

Cf. A168426 (denominators), A085737, A085738.

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

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 *)

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

A239315 Array read by antidiagonals: denominators of the core of the classical Bernoulli numbers.

Original entry on oeis.org

15, 15, 15, 105, 105, 105, 21, 105, 105, 21, 105, 105, 105, 105, 105, 15, 105, 105, 105, 105, 15, 165, 165, 1155, 231, 1155, 165, 165, 33, 165, 165, 231, 231, 165, 165, 33, 15015, 15015, 15015, 15015, 15015, 15015, 15015, 15015, 15015

Offset: 0

Paul Curtz, Mar 15 2014

We consider the autosequence A164555(n)/A027642(n) (see A190339(n)) and its difference table without the first two rows and the first two columns:
2/15,     1/15,    -1/105,   -1/21,   -1/105,     1/15,   7/165,   -5/33,...
-1/15,  -8/105,    -4/105,   4/105,    8/105,   -4/165, -32/165,...
-1/105,  4/105,     8/105,   4/105, -116/1155, -28/165,...
1/21,    4/105,    -4/105, -32/231,   -16/231,...
-1/105, -8/105, -116/1155,  16/231,...
-1/15,  -4/165,    28/165,...
7/165,  32/165,...
5/33,... etc.
This is an autosequence of the second kind.
The antidiagonals are palindromes in absolute values.
a(n) are the denominators. Multiples of 3.
Sum of odd antidiagonals: 2/15, -2/21, 2/15, -10/33, 1382/1365,... = -2*A000367(n+2)/A001897(n+2).
The sum of the even antidiagonals is A000004.
2/15, 0, -2/21,... = -4*A027641(n+4)/A027642(n+4) = -4*A164555(n)/A027642(n+4) and others.

			As a triangle:
15,
15,   15,
105, 105, 105,
21,  105, 105, 21,
105, 105, 105, 105, 105,
etc.

Cf. A085737/A085738, A168516/A168426 (autosequence), A027641, A176327/A176289, A235774, A165161/A051717(n+1).

max = 12; tb = Table[BernoulliB[n], {n, 0, max}]; td = Table[Differences[tb, n][[3 ;; -1]], {n, 2, max - 1}]; Table[td[[n - k + 1, k]] // Denominator, {n, 1, max - 3}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 11 2014 *)

A237425 Denominators of A164555(n)/A027642(n) + A198631(n)/A006519(n+1).

Original entry on oeis.org

1, 1, 6, 4, 30, 2, 42, 8, 30, 2, 66, 4, 2730, 2, 6, 16, 510, 2, 798, 4, 330, 2, 138, 8, 2730, 2, 6, 4, 870, 2, 14322, 32, 510, 2, 6, 4, 1919190, 2, 6, 8, 13530, 2, 1806, 4, 690, 2, 282, 16, 46410, 2, 66, 4, 1590, 2, 798, 8

Offset: 0

Paul Curtz, Feb 07 2014

nonn

An autosequence is a sequence which has its inverse binomial transform equal to the signed sequence. There are two possibilities. For the first kind, the main diagonal is 0's=A000004, the first two following diagonals being the same (generally not A000004). Integers example: A000045(n).
For the second kind, the main diagonal is the double of the following diagonal. Example: the companion to A000045(n) is A000032(n)=2, 1, 3, ... .
A000032(n)/2 is also a possibility. Here a(n) is the denominator of the sum of two autosequences of second kind involving (fractional) Euler and Bernoulli numbers. The corresponding fractional sequence is also an autosequence of the second kind: 2, 1, 1/6, -1/4, -1/30, 1/2, 1/42, -17/8, -1/30, 31/2, 5/66, -691/4, -691/2730,... . It could be divided by 2.

Cf. 1/n in A003506, A194767, A218289, A168516/A168426, A224270/A046161.

a[n_] := BernoulliB[n] + EulerE[n, 1]/2^IntegerExponent[n, 2]; a[0] = 2; a[1] = 1; Table[a[n] // Denominator, {n, 0, 55}] (* Jean-François Alcover, Feb 11 2014 *)

a(2n) = A002445(n). a(2n+2) = A171977(n+2).

A238235 Numerators of Euler twin numbers Et(n).

Original entry on oeis.org

1, -1, -1, -1, 1, 1, -1, -17, 17, 31, -31, -691, 691, 5461, -5461, -929569, 929569, 3202291, -3202291, -221930581, 221930581, 4722116521, -4722116521, -968383680827, 968383680827, 14717667114151, -14717667114151

Offset: 0

Paul Curtz, Feb 20 2014

sign

Et(n) = 1, -1/2, -1/2, -1/4, 1/4, 1/2, -1/2, -17/8, 17/8, 31/2, -31/2, -691/4, 691/4, 5461/2, -5461/2,... =a(n)/b(n) is mentioned in A233808.
Denominators: b(n)= 1, 2, 2, 4, 4, 2, 2, 8, 8,... = A065176(n) with 1 instead of 0.
Et(n) is the first difference of 0, followed by A198631(n)/A006519(n+1).
Et(n+2) = -1/2, -1/4, 1/4, 1/2,... is an autosequence of the second kind. Its main diagonal is the double of the following diagonal, the inverse binomial transform of Et(n+2) being Et(n+2) signed.
The denominators of the difference table of Et(n+2) are even numbers of the form 2^p. For the Bernoulli twin numbers A051716(n+1)/A051717(n+2), the denominators of the difference table, A168426(n), are multiples of 3.

Cf. A051716/A051717 (Bernoulli twin numbers).

Join[{1, -1, -1}, Table[{nu = Numerator[EulerE[2*n+1, 1]], -nu}, {n, 1, 12}]] // Flatten (* Jean-François Alcover, Feb 24 2014 *)

Binomial transform of A141424(n)/(A053644(n) with 1 instead of 0).

a(2n+3) = (-1)^n*A002425(n+2) = -a(2n+4).

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

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A239315 Array read by antidiagonals: denominators of the core of the classical Bernoulli numbers.

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A237425 Denominators of A164555(n)/A027642(n) + A198631(n)/A006519(n+1).

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A238235 Numerators of Euler twin numbers Et(n).

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