A168516 -id:A168516 - cp's OEIS frontend

A168426 Square array of denominators of a truncated array of Bernoulli twin numbers (A168516), read by antidiagonals.

3, 6, 6, 30, 15, 30, 30, 15, 15, 30, 42, 105, 105, 105, 42, 42, 21, 105, 105, 21, 42, 30, 105, 105, 105, 105, 105, 30, 30, 15, 105, 105, 105, 105, 15, 30, 66, 165, 165, 1155, 231, 1155, 165, 165, 66, 66, 33, 165, 165, 231, 231, 165, 165, 33, 66, 2730, 15015, 15015, 15015, 15015, 15015, 15015, 15015

Offset: 0

Paul Curtz, Nov 25 2009

Entries are multiples of 3.
The sequence of fractions A051716()/A051717() is a sequence of first differences of A164555()/A027642().
It can be observed (see the difference array in A190339) that A168516/A168426 is a sequence of autosequences of the second kind. - Paul Curtz, Dec 21 2016

Cf. A085738, A085737, A168516, A190339, A240581/A239315.

max = 11; 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}]] // Denominator (* Jean-François Alcover, Aug 09 2012 *)

More terms from R. J. Mathar, Jul 10 2011

A181130 Numerator of Integral_{x=0..+oo} Polylog(-n, -x)^2.

Original entry on oeis.org

1, 2, 8, 8, 32, 6112, 3712, 362624, 71706112, 3341113856, 79665268736, 1090547664896, 38770843648, 106053090598912, 5507347586961932288, 136847762542978039808, 45309996254420664320, 3447910579774800362340352

Offset: 1

Vladimir Reshetnikov, Jan 23 2011

(-1)^n*a(n) is the numerator on the main diagonal of the (truncated) array described in A168516. - Paul Curtz, Jun 20 2011

These are - up to signs - the numerators of the Bernoulli median numbers (see A212196). - Peter Luschny, May 04 2012

Peter Luschny, The computation and asymptotics of the Bernoulli numbers.

Cf. A181131 (denominator), A212196.

seq(numer((-1)^n*add(binomial(n,k)*bernoulli(n+k),k=0..n)), n=1..30); # Robert Israel, Jun 02 2015

Table[Numerator[Integrate[PolyLog[-n, -x]^2, {x, 0, Infinity}]], {n, 1, 18}]

a(n)=(-1)^n*sum(k=0,n,binomial(n,k)*bernfrac(n+k)) \\ Charles R Greathouse IV, Jun 03 2015

# uses[BernoulliMedian_list from A212196]
def A181130_list(n): return [q.numerator() for q in BernoulliMedian_list(n)]
# Peter Luschny, May 04 2012

a(n) = numerator((-1)^n/Pi^(2*n)*integral((log(t/(1-t))*log(1-1/t))^n dt,t=0,1)). - [Gerry Martens, May 25 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 *)

A256675 Denominators of the inverse binomial transform of Bernoulli(n+2).

Original entry on oeis.org

6, 6, 15, 15, 105, 21, 105, 15, 165, 33, 15015, 1365, 1365, 3, 255, 255, 33915, 399, 21945, 165, 3795, 69, 31395, 1365, 1365, 3, 435, 435, 1038345, 7161, 608685, 255, 255, 3, 959595, 959595, 959595, 3, 6765, 6765, 2036265, 903, 103845, 345, 16215, 141, 1090635

Offset: 0

Paul Curtz, Apr 07 2015

nonn

Difference table of B(n+2):
1/6,        0,  -1/30,     0,   1/42,     0, -1/30, ...
-1/6,   -1/30,   1/30,  1/42,  -1/42, -1/30,   ...
2/15,    1/15, -1/105, -1/21, -1/105,   ...
-1/15, -8/105, -4/105, 4/105,    ...
-1/105, 4/105,  8/105,   ...
1/21,   4/105,    ...
-1/105,   ...
...
a(n) is the denominator of the n-th term of the first column.
a(n+2) is the denominator of the n-th term of the third row.
See A239315(n), which is the table without the first two rows.
Inverse binomial transform: 1/6, -1/6, 2/15, -1/15, -1/105, 1/21, -1/105, -1/15, 7/165, 5/33, -2663/15015, ... .

Cf. A190339, A239315, A029765, A001897, A141459, A172087, A168516, A177735, A256595.

max = 42; bb = Table[BernoulliB[n+2], {n, 0, max}]; dd = Table[Differences[bb, n], {n, 0, max}]; dd[[All, 1]] // Denominator (* Jean-François Alcover, Apr 09 2015 *)

lista(nn) = {A = vector(nn, n, bernfrac(n+1)); for (i=1, #A-1, for(j=0,i-1,A[i+1]-=binomial(i,j)*A[j+1])); for (i=1, #A, print1(denominator(A[i]), ", "));} \\ Michel Marcus, Apr 08 2015

a(2n) = A029765(n).
a(2n+3) = A001897(n+2).
a(2n)/a(2n+1) = A177735(n).
a(2n+4)/a(2n+3) = A177735(n+3).

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).

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A168426 Square array of denominators of a truncated array of Bernoulli twin numbers (A168516), read by antidiagonals.

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A181130 Numerator of Integral_{x=0..+oo} Polylog(-n, -x)^2.

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

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A256675 Denominators of the inverse binomial transform of Bernoulli(n+2).

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

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