A181130 -id:A181130 - cp's OEIS frontend

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

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

A181131 Denominator of Integral_{x=0..+oo} Polylog(-n, -x)^2 for n > 0, with a(0) = 1.

Original entry on oeis.org

1, 3, 15, 105, 105, 231, 15015, 2145, 36465, 969969, 4849845, 10140585, 10140585, 22287, 3231615, 7713865005, 7713865005, 90751353, 218257003965, 1641030105, 67282234305, 368217318651, 1841086593255, 3762220429695, 63957747304815, 1546231253523

Offset: 0

Vladimir Reshetnikov, Jan 23 2011

These are the denominators of the Bernoulli median numbers (see A212196). - Peter Luschny, May 04 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A181130, A212196.

seq(denom(add(binomial(n,k)*bernoulli(n+k),k=0..n)), n=0..100); # Robert Israel, Jun 02 2015

Table[Denominator[Integrate[PolyLog[-n, -x]^2, {x, 0, Infinity}]], {n, 1, 18}]
max = 25; t[0] = Table[BernoulliB[n], {n, 0, 2*max}]; t[n_] := Differences[t[0], n]; a[n_] := t[n][[n + 1]] // Denominator; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Jul 25 2013, after Peter Luschny *)

a(n)=denominator(-subst(intformal(polylog(-n,-x)^2),'x,0)) \\ Charles R Greathouse IV, Jul 21 2014

# uses[BernoulliMedian_list from A212196]
def A181131_list(n):
    return [denominator(q) for q in BernoulliMedian_list(n)]
# Peter Luschny, May 04 2012

a(n) = denominator((-1)^n/Pi^(2*n)*integral((log(t/(1-t))*log(1-1/t))^n dt,t=0,1)). - [Gerry Martens, May 25 2011]

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

Offset set to 0, a(0) and a(19)..a(25) added by Peter Luschny, May 04 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

A195240 Numerators of the second differences of the sequence of fractions (-1)^(n+1)*A176618(n)/A172031(n).

Original entry on oeis.org

0, 1, 1, 7, 8, 11, 10, 7, 8, 19, 14, 337, 1028, 5, -2, -1681, 1936, 22133, -21734, -87223, 87388, 427291, -427222, -118181363, 118182728, 4276553, -4276550, -11874730297, 11874730732, 4307920641583

Offset: 0

Paul Curtz, Sep 13 2011

The array of (-1)^n*A176328(n)/A176591(n) and its first, second, etc. differences in subsequence rows starts as follows:
0,     1,     2,   19/6,   14/3, 199/30,    137/15, ... (-1)^n * A176328(n)/A176591(n),
1,     1,   7/6,    3/2,  59/30,    5/2,    127/42, ... see A176328,
0,   1/6,   1/3,   7/15,   8/15,  11/21,     10/21, ...
1/6, 1/6,  2/15,   1/15, -1/105,  -1/21,    -1/105, ... see A190339
0, -1/30, -1/15, -8/105, -4/105,  4/105, -116/1155, ...
The numerators in the 3rd row, 0, 1/6, 1/3, 7/15, 8/15, 11/21, 10/21, 7/15, 8/15, 19/33, 14/33, 337/1365, 1028/1365, 5/3, -2/3, -1681/255, 1936/255, ... define the current sequence.
The associated denominators are 1, 6 and followed by 3, 15, 15 etc as provided in A172087.
The second column of the array, 1, 1, 1/6, 1/6, -1/30, -1/30, ... contains doubled A000367(n)/A002445(n). These are related to A176150, A176144, and A176184.
In the first subdiagonal of the array we see 1, 1/6, 2/15, -8/150, 8/105, -32/321, 6112/15015, -3712/2145 , ... continued as given by A181130 and A181131.

Vincenzo Librandi, Table of n, a(n) for n = 0..300

read("transforms") ;
evb := [0, 1, 0, seq(bernoulli(n), n=2..30)] ;
ievb := BINOMIALi(evb) ;
[seq((-1)^n*op(n,ievb),n=1..nops(ievb))] ;
DIFF(%) ;
DIFF(%) ;
apply(numer,%) ; # R. J. Mathar, Sep 20 2011

evb = Join[{0, 1, 0}, Table[BernoulliB[n], {n, 2, 32}]]; ievb = Table[ Sum[Binomial[n, k]*evb[[k+1]], {k, 0, n}], {n, 0, Length[evb]-3}]; Differences[ievb, 2] // Numerator (* Jean-François Alcover, Sep 09 2013, after R. J. Mathar *)

a(2*n+1) + a(2*n+2) = A172087(2*n+2) = A172087(2*n+3), n >= 1.

A191972 The numerators of 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.

Original entry on oeis.org

1, -1, 1, -4, 4, -16, 3056, -1856, 181312, -35853056, 1670556928, -39832634368, 545273832448, -19385421824, 53026545299456, -2753673793480966144, 68423881271489019904, -22654998127210332160

Offset: 0

Paul Curtz, Jun 20 2011

For the denominators of T(n, n+1) see A190339, where detailed information can be found.

			T(n,n+1) = [1/2, -1/6, 1/15 , -4/105, 4/105, -16/231, 3056/15015, -1856/2145, 181312/36465, ...]

Cf. A191302, A191754, A181131.

nmax:=20: 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: for n from 0 to nmax do seq(T(n,m),m=0..mmax) od: seq(numer(T(n,n+1)),n=0..nmax-1); # Johannes W. Meijer, Jun 30 2011

nmax = 17; b[n_] := BernoulliB[n]; b[1] = 1/2; bb = Table[b[n], {n, 0, 2*nmax+1}]; dd = Table[Differences[bb, n], {n, 1, nmax }]; a[0] = 1; a[n_] := dd[[n, n+2]] // Numerator; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Oct 02 2012 *)

T(n, n+1) = T(n, n)/2.

a(n+2) = (-1)^n*A181130(n+2)/2.

Thanks to R. J. Mathar by Paul Curtz, Jun 20 2011

Edited by Johannes W. Meijer, Jun 30 2011

cp's OEIS Frontend

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

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A181131 Denominator of Integral_{x=0..+oo} Polylog(-n, -x)^2 for n > 0, with a(0) = 1.

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A212196 Numerators of the Bernoulli median numbers.

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A195240 Numerators of the second differences of the sequence of fractions (-1)^(n+1)*A176618(n)/A172031(n).

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A191972 The numerators of 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.

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