A085738 -id:A085738 - cp's OEIS frontend

A168464 a(n) = A085738(n) - A085737(n).

0, 1, 1, 5, 2, 5, 1, 5, 5, 1, 31, 29, 13, 29, 31, 1, 31, 14, 14, 31, 1, 41, 43, 106, 97, 106, 43, 41, 1, 41, 22, 101, 101, 22, 41, 1, 31, 29, 106, 109, 97, 109, 106, 29, 31, 1, 31, 14, 113, 101, 101, 113, 14, 31, 1, 61, 71, 158, 161, 1271, 199, 1271, 161, 158, 71, 61, 1, 61, 38

Offset: 0

Paul Curtz, Nov 26 2009

The corresponding sum is in A168140.

The sequence contains negative numbers starting in row T(15,.) of the triangle.

			As a triangle, sequence begins:
   0;
   1,  1;
   5,  2,  5;
   1,  5,  5,  1;
  31, 29, 13, 29, 31;
   1, 31, 14, 14, 31, 1;
  ...

Cf. A085737, A085738, A168140.

T := proc(n,k) option remember; if k = 0 then (-1)^n*bernoulli(n) ; elif k > n/2 then procname(n,n-k) ; else procname(n-1,k-1)-procname(n,k-1) ; end if; end proc:
A085737 := proc(n,k) numer(T(n,k)) ; end proc:
A085738 := proc(n,k) denom(T(n,k)) ; end proc:
for n from 0 to 15 do for k from 0 to n do printf("%d,",A085738(n,k)-A085737(n,k)) ; end do: end do: # R. J. Mathar, Mar 21 2010

t(n,k) = if (k==0, (-1)^n*bernfrac(n), t(n-1, k-1) - t(n, k-1));
T(n,k) = my(tnk=t(n,k)); denominator(tnk) - numerator(tnk);
tabl(nn) = for (n=0, nn, for (k= 0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Feb 01 2019

Offset set to zero, sequence extended by R. J. Mathar, Mar 21 2010

Keywords sign and tabl from Michel Marcus, Feb 01 2019

A190339 The denominators of the subdiagonal in the difference table of the Bernoulli numbers.

Original entry on oeis.org

2, 6, 15, 105, 105, 231, 15015, 2145, 36465, 969969, 4849845, 10140585, 10140585, 22287, 3231615, 7713865005, 7713865005, 90751353, 218257003965, 1641030105, 67282234305, 368217318651, 1841086593255

Offset: 0

Paul Curtz, May 09 2011

Apparently a(n) = A181131(n) for n>=2 (checked numerically up to n=640). - R. J. Mathar, Aug 25 2025
The denominators of the T(n, n+1) with T(0, m) = A164555(m)/A027642(m) and T(n, m) = T(n-1, m+1) - T(n-1, m), n >= 1, m >= 0. For the numerators of the T(n, n+1) see A191972.
The T(n, m) are defined by A164555(n)/A027642(n) and its successive differences, see the formulas.
Reading the array T(n, m), see the examples, by its antidiagonals leads to A085737(n)/A085738(n).
A164555(n)/A027642(n) is an autosequence (eigensequence whose inverse binomial transform is the sequence signed) of the second kind; the main diagonal T(n, n) is twice the first upper diagonal T(n, n+1).
We can get the Bernoulli numbers from the T(n, n+1) in an original way, see A192456/A191302.
Also the denominators of T(n, n+1) of the table defined by A085737(n)/A085738(n), the upper diagonal, called the median Bernoulli numbers by Chen. As such, Chen proved that a(n) is even only for n=0 and n=1 and that a(n) are squarefree numbers. (see Chen link). - Michel Marcus, Feb 01 2013
The sum of the antidiagonals of T(n,m) is 1 in the first antidiagonal, otherwise 0. Paul Curtz, Feb 03 2015

			The first few rows of the T(n, m) array (difference table of the Bernoulli numbers) are:
1,       1/2,     1/6,      0,     -1/30,         0,        1/42,
-1/2,   -1/3,    -1/6,  -1/30,      1/30,      1/42,       -1/42,
1/6,     1/6,    2/15,   1/15,    -1/105,     -1/21,      -1/105,
0,     -1/30,   -1/15, -8/105,    -4/105,     4/105,       8/105,
-1/30, -1/30,  -1/105,  4/105,     8/105,     4/105,   -116/1155,
0,      1/42,    1/21,  4/105,    -4/105,   -32/231,     -16/231,
1/42,   1/42,  -1/105, -8/105, -116/1155,    16/231,  6112/15015,

Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187.

G. C. Greubel, Table of n, a(n) for n = 0..450
Kwang-Wu Chen, A summation on Bernoulli numbers, Journal of Number Theory, Volume 111, Issue 2, April 2005, Pages 372-391.
Peter Luschny, Computation and asymptotics of the Bernoulli numbers

T := proc(n,m)
    option remember;
    if n < 0 or m < 0 then
        0 ;
    elif n = 0 then
        if m = 1 then
            -bernoulli(m) ;
        else
            bernoulli(m) ;
        end if;
    else
        procname(n-1,m+1)-procname(n-1,m) ;
    end if;
end proc:
A190339 := proc(n)
    denom( T(n+1,n)) ;
end proc: # R. J. Mathar, Apr 25 2013

nmax = 23; b[n_] := BernoulliB[n]; b[1]=1/2; bb = Table[b[n], {n, 0, 2*nmax-1}]; diff = Table[Differences[bb, n], {n, 1, nmax}]; Diagonal[diff] // Denominator (* Jean-François Alcover, Aug 08 2012 *)

def A190339_list(n) :
    T = matrix(QQ, 2*n+1)
    for m in (0..2*n) :
        T[0,m] = bernoulli_polynomial(1,m)
        for k in range(m-1,-1,-1) :
            T[m-k,k] = T[m-k-1,k+1] - T[m-k-1,k]
    for m in (0..n-1) : print([T[m,k] for k in (0..n-1)])
    return [denominator(T[k,k+1]) for k in (0..n-1)]
A190339_list(7) # Also prints the table as displayed in EXAMPLE. Peter Luschny, Jun 21 2012

T(0, m) = A164555(m)/A027642(m) and T(n, m) = T(n-1, m+1) - T(n-1, m), n >= 1, m >= 0.
T(1, m) = A051716(m+1)/A051717(m+1);
T(n, n) = 2*T(n, n+1).
T(n+1, n+1) = (-1)^(1+n)*A181130(n+1)/A181131(n+1). - R. J. Mathar, Jun 18 2011
a(n) = A141044(n)*A181131(n). - Paul Curtz, Apr 21 2013

Edited and Maple program added by Johannes W. Meijer, Jun 29 2011, Jun 30 2011

New name from Peter Luschny, Jun 21 2012

A212196 Numerators of the Bernoulli median numbers.

Original entry on oeis.org

1, -1, 2, -8, 8, -32, 6112, -3712, 362624, -71706112, 3341113856, -79665268736, 1090547664896, -38770843648, 106053090598912, -5507347586961932288, 136847762542978039808, -45309996254420664320, 3447910579774800362340352, -916174777198089643491328

Offset: 0

Peter Luschny, May 04 2012

The Bernoulli median numbers are the numbers in the median (central) column of the difference table of the Bernoulli numbers.
The Sage script below is based on L. Seidel's algorithm and does not make use of a library function for the Bernoulli numbers; in fact it generates the Bernoulli numbers on the fly.
A181130 is an unsigned version with offset 1. A181131 are the denominators of the Bernoulli median numbers.

			The difference table of the Bernoulli numbers, [m] the Bernoulli median numbers.
     [1]
    1/2,  -1/2
    1/6,[-1/3],  1/6
      0,  -1/6,   1/6,       0
  -1/30, -1/30,[2/15],   -1/30,    -1/30
      0,  1/30,  1/15,   -1/15,    -1/30,         0
   1/42,  1/42,-1/105,[-8/105],   -1/105,      1/42,      1/42
      0, -1/42, -1/21,  -4/105,    4/105,      1/21,      1/42,      0
  -1/30, -1/30,-1/105,   4/105,  [8/105],     4/105,    -1/105,  -1/30, -1/30
      0,  1/30,  1/15,   8/105,    4/105,    -4/105,    -8/105,  -1/15, -1/30, 0
   5/66,  5/66, 7/165,  -4/165,-116/1155, [-32/231], -116/1155, -4/165, 7/165, ..
.
Integral_{x=0..1} 1 = 1
Integral_{x=0..1} (-1)^1*x^2 = -1/3
Integral_{x=0..1} (-1)^2*(2*x^2 - x)^2 = 2/15
Integral_{x=0..1} (-1)^3*(6*x^3 - 6*x^2 + x)^2 = -8/105,
Integral_{x=0..1} (-1)^4*(24*x^4 - 36*x^3 + 14*x^2 - x)^2 = 8/105
Integral_{x=0..1} (-1)^5*(120*x^5 - 240*x^4 + 150*x^3 - 30*x^2 + x)^2 = -32/231,
...
Integral_{x=0..1} (-1)^n*(Sum_{k=0..n} Stirling2(n,k)*k!*(-x)^k)^2 = BernoulliMedian(n).
Compare A164555. - _Peter Luschny_, Aug 13 2017

Peter Luschny, The computation and asymptotics of the Bernoulli numbers.
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187.

Cf. A164555, A181130, A181131, A085737, A085738.

max = 19; t[0] = Table[ BernoulliB[n], {n, 0, 2*max}]; t[n_] := Differences[t[0], n]; a[1] = -1; a[n_] := t[n][[n + 1]] // Numerator; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Jun 26 2013 *)

def BernoulliMedian_list(n) :
    def T(S, a) :
        R = [a]
        for s in S :
            a -= s
            R.append(a)
        return R
    def M(A, p) :
        R = T(A,0)
        S = add(r for r in R)
        return -S / (2*p+3)
    R = [1]; A = [1/2, -1/2]
    for k in (0..n-2) :
        A = T(A, M(A,k))
        R.append(A[k+1])
        A = T(A,0)
    return R
def A212196_list(n): return [numerator(b) for b in BernoulliMedian_list(n)]

a(n) = numerator(Sum_{k=0..n} C(n,k)*Bernoulli(n+k)). - Vladimir Kruchinin, Apr 06 2015

A085737 Numerators in triangle formed from Bernoulli numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, 1, 1, 0, -1, 1, 2, 1, -1, 0, -1, 1, 1, -1, 0, 1, -1, -1, 8, -1, -1, 1, 0, 1, -1, 4, 4, -1, 1, 0, -1, 1, -1, -4, 8, -4, -1, 1, -1, 0, -1, 1, -8, 4, 4, -8, 1, -1, 0, 5, -5, 7, 4, -116, 32, -116, 4, 7, -5, 5, 0, 5, -5, 32, -28, 16, 16, -28, 32, -5, 5, 0

Offset: 0

N. J. A. Sloane, following a suggestion of J. H. Conway, Jul 23 2003

Triangle is determined by rules 0) the top number is 1; 1) each number is the sum of the two below it; 2) it is left-right symmetric; 3) the numbers in each of the border rows, after the first 3, are alternately 0.

Up to signs this is the difference table of the Bernoulli numbers (see A212196). The Sage script below is based on L. Seidel's algorithm and does not make use of a library function for the Bernoulli numbers; in fact it generates the Bernoulli numbers on the fly. - Peter Luschny, May 04 2012

			Triangle of fractions begins
    1;
   1/2,   1/2;
   1/6,   1/3,   1/6;
    0,    1/6,   1/6,     0;
  -1/30,  1/30,  2/15,   1/30,  -1/30;
    0,   -1/30,  1/15,   1/15,  -1/30,    0;
   1/42, -1/42, -1/105,  8/105, -1/105, -1/42,   1/42;
    0,    1/42, -1/21,   4/105,  4/105, -1/21,   1/42,   0;
  -1/30,  1/30, -1/105, -4/105,  8/105, -4/105, -1/105, 1/30, -1/30;

Fabien Lange and Michel Grabisch, The interaction transform for functions on lattices Discrete Math. 309 (2009), no. 12, 4037-4048. [From _N. J. A. Sloane_, Nov 26 2011]
Peter Luschny, The computation and asymptotics of the Bernoulli numbers.
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [_Peter Luschny_, May 04 2012]

Cf. A085738, A212196. See A051714/A051715 for another triangle that generates the Bernoulli numbers.

nmax:=11; for n from 0 to nmax do T(n, 0):= (-1)^n*bernoulli(n) od: for n from 1 to nmax do for k from 1 to n do  T(n, k) := T(n-1, k-1) - T(n, k-1) od: od: for n from 0 to nmax do seq(T(n, k), k=0..n) od: seq(seq(numer(T(n, k)), k=0..n), n=0..nmax);  # Johannes W. Meijer, Jun 29 2011, revised Nov 25 2012

t[n_, 0] := (-1)^n*BernoulliB[n]; t[n_, k_] := t[n, k] = t[n-1, k-1] - t[n, k-1]; Table[t[n, k] // Numerator, {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 07 2014 *)

def BernoulliDifferenceTable(n) :
    def T(S, a) :
        R = [a]
        for s in S :
            a -= s
            R.append(a)
        return R
    def M(A, p) :
        R = T(A,0)
        S = add(r for r in R)
        return -S / (2*p+3)
    R = [1/1]
    A = [1/2,-1/2]; R.extend(A)
    for k in (0..n-2) :
        A = T(A,M(A,k)); R.extend(A)
        A = T(A,0); R.extend(A)
    return R
def A085737_list(n) : return [numerator(q) for q in BernoulliDifferenceTable(n)]
# Peter Luschny, May 04 2012

T(n, 0) = (-1)^n*Bernoulli(n), T(n, k) = T(n-1, k-1) - T(n, k-1) for k=1..n.

T(n,k) = Sum_{j=0..k} binomial(k,j)*Bernoulli(n-j). [Lange and Grabisch]

Sign flipped in formula by Johannes W. Meijer, Jun 29 2011

A191302 Denominators in triangle that leads to the Bernoulli numbers.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 2, 3, 15, 2, 6, 3, 2, 1, 5, 105, 2, 6, 15, 15, 2, 3, 3, 105, 105, 2, 2, 5, 7, 35, 2, 3, 3, 21, 21, 231, 2, 6, 15, 15, 21, 21, 2, 1, 5, 15, 1, 77, 15015, 2, 6, 3, 35, 15, 33, 1155

Offset: 0

Paul Curtz, May 30 2011

For the definition of the ASPEC array coefficients see the formulas; see also A029635 (Lucas triangle), A097207 and A191662 (k-dimensional square pyramidal numbers).
The antidiagonal row sums of the ASPEC array equal A042950(n) and A098011(n+3).
The coefficients of the T(n,m) array are defined in A190339. We define the coefficients of the SBD array with the aid of the T(n,n+1), see the formulas and the examples.
Multiplication of the coefficients in the rows of the ASPEC array with the coefficients in the columns of the SBD array leads to the coefficients of the BSPEC triangle, see the formulas. The BSPEC triangle can be looked upon as a spectrum for the Bernoulli numbers.
The row sums of the BSPEC triangle give the Bernoulli numbers A164555(n)/A027642(n).
For the numerators of the BSPEC triangle coefficients see A192456.

			The first few rows of the array ASPEC array:
  2, 1,  1,  1,   1,   1,    1,
  2, 3,  4,  5,   6,   7,    8,
  2, 5,  9, 14,  20,  27,   35,
  2, 7, 16, 30,  50,  77,  112,
  2, 9, 25, 55, 105, 182,  294,
The first few T(n,n+1) = T(n,n)/2 coefficients:
1/2, -1/6, 1/15, -4/105, 4/105, -16/231, 3056/15015, ...
The first few rows of the SBD array:
  1/2,   0,   0,     0
  1/2,   0,   0,     0
  1/2, -1/6,  0,     0
  1/2, -1/6,  0,     0
  1/2, -1/6, 1/15,   0
  1/2, -1/6, 1/15,   0
  1/2, -1/6, 1/15, -4/105
  1/2, -1/6, 1/15, -4/105
The first few rows of the BSPEC triangle:
  B(0) =   1   = 1/1
  B(1) =  1/2  = 1/2
  B(2) =  1/6  = 1/2 - 1/3
  B(3) =   0   = 1/2 - 1/2
  B(4) = -1/30 = 1/2 - 2/3 +  2/15
  B(5) =   0   = 1/2 - 5/6 +  1/3
  B(6) =  1/42 = 1/2 - 1/1 +  3/5  - 8/105
  B(7) =   0   = 1/2 - 7/6 + 14/15 - 4/15

Cf. A028246 (Worpitzky), A085737/A085738 (Conway-Sloane) and A051714/A051715 (Akiyama-Tanigawa) for other triangles that lead to the Bernoulli numbers. - Johannes W. Meijer, Jul 02 2011

nmax:=13: mmax:=nmax:
A164555:=proc(n): if n=1 then 1 else numer(bernoulli(n)) fi: end:
A027642:=proc(n): if n=1 then 2 else denom(bernoulli(n)) fi: end:
for m from 0 to 2*mmax do T(0,m):=A164555(m)/A027642(m) od:
for n from 1 to nmax do for m from 0 to 2*mmax do T(n,m):=T(n-1,m+1)-T(n-1,m) od: od:
seq(T(n,n+1),n=0..nmax):
for n from 0 to nmax do ASPEC(n,0):=2: for m from 1 to mmax do ASPEC(n,m):= (2*n+m)*binomial(n+m-1,m-1)/m od: od:
for n from 0 to nmax do seq(ASPEC(n,m),m=0..mmax) od:
for n from 0 to nmax do for m from 0 to 2*mmax do SBD(n,m):=0 od: od:
for m from 0 to mmax do for n from 2*m to nmax do SBD(n,m):= T(m,m+1) od: od:
for n from 0 to nmax do seq(SBD(n,m), m= 0..mmax/2) od:
for n from 0 to nmax do BSPEC(n,2) := SBD(n,2)*ASPEC(2,n-4) od:
for m from 0 to mmax do for n from 0 to nmax do BSPEC(n,m) := SBD(n,m)*ASPEC(m,n-2*m) od: od:
for n from 0 to nmax do seq(BSPEC(n,m), m=0..mmax/2) od:
seq(add(BSPEC(n, k), k=0..floor(n/2)) ,n=0..nmax):
Tx:=0:
for n from 0 to nmax do for m from 0 to floor(n/2) do a(Tx):= denom(BSPEC(n,m)): Tx:=Tx+1: od: od:
seq(a(n),n=0..Tx-1); # Johannes W. Meijer, Jul 02 2011

(* a=ASPEC, b=BSPEC *) nmax = 13; a[n_, 0] = 2; a[n_, m_] := (2n+m)*Binomial[n+m-1, m-1]/m; b[n_] := BernoulliB[n]; b[1]=1/2; bb = Table[b[n], {n, 0, nmax}]; diff = Table[ Differences[bb, n], {n, 1, nmax}]; dd = Diagonal[diff]; sbd[n_, m_] := If[n >= 2m, -dd[[m+1]], 0]; b[n_, m_] := sbd[n, m]*a[m, n-2m]; Table[b[n, m], {n, 0, nmax}, {m, 0, Floor[n/2]}] // Flatten // Denominator (* Jean-François Alcover_, Aug 09 2012 *)

ASPEC(n, 0) = 2 and ASPEC(n, m) = (2*n+m)*binomial(n+m-1, m-1)/m, n >= 0, m >= 1.
ASPEC(n, m) = ASPEC(n-1, m) + ASPEC(n, m-1), n >= 1, m >= 1, with ASPEC(n, 0) = 2, n >= 0, and ASPEC(0,m) = 1, m >= 1.
SBD(n, m) = T(m, m+1), n >= 2*m; see A190339 for the definition of the T(n, m).
BSPEC(n, m) = SBD(n, m)*ASPEC(m, n-2*m)
Sum_{k=0..floor(n/2)} BSPEC(n, k) = A164555(n)/A027642(n).

Edited, Maple program and crossrefs added by Johannes W. Meijer, Jul 02 2011

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

Original entry on oeis.org

-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

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

Original entry on oeis.org

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

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 *)

A168140 Sum of n-th numerator and n-th denominator in triangle formed from Bernoulli numbers.

Original entry on oeis.org

2, 3, 3, 7, 4, 7, 1, 7, 7, 1, 29, 31, 17, 31, 29, 1, 29, 16, 16, 29, 1, 43, 41, 104, 113, 104, 41, 43, 1, 43, 20, 109, 109, 20, 43, 1, 29, 31, 104, 101, 113, 101, 104, 31, 29, 1, 29, 16, 97, 109, 109, 97, 16, 29, 1, 71, 61, 172, 169, 1039, 263, 1039, 169, 172, 61, 71, 1, 71, 28, 197, 137, 247, 247, 137, 197, 28, 71, 1

Offset: 0

Juri-Stepan Gerasimov, Nov 19 2009

Terms become negative later in the sequence. - R. J. Mathar, Apr 14 2010

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A085737, A085738, A162298, A162299.

b := proc(n, k) option remember; if k = 0 then (-1)^n*bernoulli(n) ; elif k > n/2 then procname(n, n-k) ; else procname(n-1, k-1)-procname(n, k-1) ; end if; end proc: T := proc(n, k) numer(b(n, k))+denom(b(n, k)) ; end proc: seq(seq(T(n,k), k=0..n), n=0..10); # R. J. Mathar, Apr 14 2010

b[n_, 0] := (-1)^n BernoulliB[n];
b[n_, k_] := b[n, k] = b[n - 1, k - 1] - b[n, k - 1];
T[n_, k_] := Numerator[b[n, k]] + Denominator[b[n, k]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 21 2020 *)

a(n) = A085737(n) + A085738(n).

A 109 duplicated, 1049 replaced with 1039 by R. J. Mathar, Apr 14 2010

A227577 Square array read by antidiagonals, A(n,k) the numerators of the elements of the difference table of the Euler polynomials evaluated at x=1, for n>=0, k>=0.

Original entry on oeis.org

1, -1, 1, 0, -1, 0, 1, 1, -1, -1, 0, 1, 1, 1, 0, -1, -1, -1, 1, 1, 1, 0, -1, -1, -5, -1, -1, 0, 17, 17, 13, 5, -5, -13, -17, -17, 0, 17, 17, 47, 13, 47, 17, 17, 0, -31, -31, -107, -73, -13, 13, 73, 107, 31, 31, 0, -31, -31, -355

Offset: 0

Paul Curtz, Jul 16 2013

sign

The difference table of the Euler polynomials evaluated at x=1:
    1,   1/2,     0,    -1/4,     0,     1/2,      0,      -17/8, ...
  -1/2, -1/2,   -1/4,    1/4,    1/2,   -1/2,   -17/8,      17/8, ...
    0,   1/4,    1/2,    1/4;    -1,   -13/8,    17/4,     107/8, ...
   1/4,  1/4,   -1/4,   -5/4,   -5/8,   47/8,    73/8,    -355/8, ...
    0,  -1/2,    -1,     5/8    13/2,   13/4,  -107/2,    -655/8, ...
  -1/2, -1/2,   13/8,   47/8,  -13/4, -227/4,  -227/8,    5687/8, ...
    0,  17/8,   17/4,  -73/8, -107/2,  227/8,  2957/4,    2957/8, ...
  17/8, 17/8, -107/8, -355/8,  655/8, 5687/8, -2957/8, -107125/8, ...
To compute the difference table, take
    1,   1/2;
  -1/2;
The next term is always half of the sum of the antidiagonals. Hence (-1/2 + 1/2 = 0)
    1,   1/2,   0;
  -1/2, -1/2;
    0;
The first column (inverse binomial transform) lists the numbers (1, -1/2, 0, 1/4, ..., not in the OEIS; corresponds to A027641/A027642). See A209308 and A060096.
A198631(n)/A006519(n+1) is an autosequence. See A181722.
Note the main diagonal: 1, -1/2, 1/2, -5/4, 13/2, -227/4, 2957/4, -107125/8, .... (See A212196/A181131.)
This twice the first upper diagonal. The autosequence is of the second kind.
From 0, -1, the algorithm gives A226158(n), full Genocchi numbers, autosequence of the first kind.
The difference table of the Bernoulli polynomials evaluated at x=1 is (apart from signs) A085737/A085738 and its analysis by Ludwig Seidel was discussed in the Luschny link. - Peter Luschny, Jul 18 2013

			Read by antidiagonals:
    1;
  -1/2,  1/2;
    0,  -1/2,   0;
   1/4,  1/4, -1/4, -1/4;
    0,   1/4,  1/2,  1/4,   0;
  -1/2, -1/2, -1/4,  1/4,  1/2,  1/2;
    0,  -1/2, - 1,  -5/4,  -1,  -1/2,   0;
  ...
Row sums: 1, 0, -1/2, 0, 1, 0, -17/4, 0, ... = 2*A198631(n+1)/A006519(n+2).
Denominators: 1, 1, 2, 1, 1, 1, 4, 1, ... = A160467(n+2)?

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

Cf. A164555/A027642 in A190339.

DifferenceTableEulerPolynomials := proc(n) local A,m,k,x;
A := array(0..n,0..n); x := 1;
for m from 0 to n do for k from 0 to n do A[m,k]:= 0 od od;
for m from 0 to n do A[m,0] := euler(m,x);
   for k from m-1 by -1 to 0 do
      A[k,m-k] := A[k+1,m-k-1] - A[k,m-k-1] od od;
LinearAlgebra[Transpose](convert(A, Matrix)) end:
DifferenceTableEulerPolynomials(7);  # Peter Luschny, Jul 18 2013

t[0, 0] = 1; t[0, k_] := EulerE[k, 1]; t[n_, 0] := -t[0, n]; t[n_, k_] := t[n, k] = t[n-1, k+1] - t[n-1, k]; Table[t[n-k, k] // Numerator, {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 18 2013 *)

def DifferenceTableEulerPolynomialsEvaluatedAt1(n) :
    @CachedFunction
    def ep1(n):          # Euler polynomial at x=1
        if n < 2: return 1 - n/2
        s = add(binomial(n,k)*ep1(k) for k in (0..n-1))
        return 1 - s/2
    T = matrix(QQ, n)
    for m in range(n) :  # Compute difference table
        T[m,0] = ep1(m)
        for k in range(m-1,-1,-1) :
            T[k,m-k] = T[k+1,m-k-1] - T[k,m-k-1]
    return T
def A227577_list(m):
    D = DifferenceTableEulerPolynomialsEvaluatedAt1(m)
    return [D[k,n-k].numerator() for n in range(m) for k in (0..n)]
A227577_list(12)  # Peter Luschny, Jul 18 2013

Corrected by Jean-François Alcover, Jul 17 2013

cp's OEIS Frontend

A168464 a(n) = A085738(n) - A085737(n).

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A190339 The denominators of the subdiagonal in the difference table of the Bernoulli numbers.

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A212196 Numerators of the Bernoulli median numbers.

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A085737 Numerators in triangle formed from Bernoulli numbers.

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A191302 Denominators in triangle that leads to the Bernoulli numbers.

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

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