A181131 -id:A181131 - cp's OEIS frontend

A240776 Define a square array B(m,n) (m>=0, n>=0) by B(n, n) = A212196(n)/A181131(n), B(n, n+1) = -A212196(n)/A181131(n), B(m, n) = B(m, n-1) + B(m+1, n-1); a(n) = numerator of B(0,n).

1, -1, -4, -1, -8, -1, -8, -3, -8, 1, 104, -41, -920, 1767, 20168, -8317, -2022392, 869807, 291391192, -129169263, -2759924456, 250158593, 146772324808, -67632514765, -10164962436952

Offset: 0

Paul Curtz, Apr 12 2014

The array B(m,n) begins:
1,          -1,  -4/3,     -1,   -8/15,   -1/5, -8/105,...
-2,       -1/3,   1/3,   7/15,     1/3, 13/105,...
5/3,       2/3,  2/15,  -2/15, -22/105,...
-1,      -8/15, -4/15, -8/105,...
7/15,     4/15,  4/21,...
-1/5,   -8/105,...
13/105,...
etc.
B(0, n) = 1, -1, -4/3, -1, -8/15, -1/5, -8/105, -3/35, -8/105, 1/35, 104/1155, ... = a(n)/b(n).
The main diagonal is A212196(n)/A181131(n).
The first upper diagonal is -A212196(n)/A181131(n).

max = 12; t[0] = Table[BernoulliB[n], {n, 0, 2*max}]; t[n_] := t[n] = Differences[t[0], n]; B1[1, 1] = -1/3; B1[n_, n_] := t[n][[n+1]]; B1[m_, n_] /; n == m+1 := B1[m, n] = -B1[m, m]; B1[m_?NonNegative, n_?NonNegative] := B1[m, n] = B1[m, n-1] + B1[m+1, n-1]; B1[, ] = 0; Table[B1[0, n] // Numerator, {n, 0, 2*max}] (* Jean-François Alcover, Apr 14 2014 *)

More terms from Jean-François Alcover, Apr 14 2014

Edited by N. J. A. Sloane, May 21 2014

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

A291447 Triangle read by rows, numerators of coefficients (in rising powers) of rational polynomials P(n, x) such that Integral_{x=0..1} P'(n, x) = BernoulliMedian(n).

Original entry on oeis.org

0, 1, 0, 0, 0, 1, 0, 0, 0, 1, -1, 4, 0, 0, 0, 1, -3, 48, -12, 36, 0, 0, 0, 1, -7, 268, -176, 1968, -216, 64, 0, 0, 0, 1, -15, 240, -1580, 37140, -9900, 10400, -5760, 14400, 0, 0, 0, 1, -31, 4924, -11680, 488640, -238680, 496320, -639360, 5486400, -216000, 518400

Offset: 0

Peter Luschny, Aug 24 2017

The Bernoulli median numbers are A212196/A181131. See A290694 for further comments.

			Triangle starts:
[0, 1]
[0, 0, 0, 1]
[0, 0, 0, 1, -1, 4]
[0, 0, 0, 1, -3, 48, -12, 36]
[0, 0, 0, 1, -7, 268, -176, 1968, -216, 64]
[0, 0, 0, 1, -15, 240, -1580, 37140, -9900, 10400, -5760, 14400]
The first few polynomials are:
P_0(x) = x.
P_1(x) = (1/3)*x^3.
P_2(x) = (4/5)*x^5 - x^4 + (1/3)*x^3.
P_3(x) = (36/7)*x^7 - 12*x^6 + (48/5)*x^5 - 3*x^4 + (1/3)*x^3.
P_4(x) = 64*x^9 - 216*x^8 + (1968/7)*x^7 - 176*x^6 + (268/5)*x^5 - 7*x^4 +(1/3)*x^3.
Evaluated at x = 1 this gives a decomposition of the Bernoulli median numbers:
BM(0) = 1     =    1.
BM(1) = 1/3   =  1/3.
BM(2) = 2/15  =  4/5 -   1 +    1/3.
BM(3) = 8/105 = 36/7 -  12 +   48/5 -   3 +   1/3.
BM(4) = 8/105 =   64 - 216 + 1968/7 - 176 + 268/5 - 7 + 1/3.

Peter Luschny, Illustrating A291447

Cf. A164555/A027642, A212196/A181131, A291449/A291450, A290694/A290695, A291447/A291448.

# The function BG_row is defined in A290694.
seq(BG_row(2, n, "num", "val"), n=0..12);        # A212196
seq(BG_row(2, n, "den", "val"), n=0..12);        # A181131
seq(print(BG_row(2, n, "num", "poly")), n=0..7); # A291447 (this seq.)
seq(print(BG_row(2, n, "den", "poly")), n=0..9); # A291448

T[n_] := Integrate[Sum[(-1)^(n-j+1) StirlingS2[n, j] j! x^j, {j, 0, n}]^2, x];
Trow[n_] := CoefficientList[T[n], x] // Numerator;
Table[Trow[r], {r, 0, 6}] // Flatten

T(n,k) = Numerator([x^k] Integral(Sum_{j=0..n}(-1)^(n-j)*Stirling2(n,j)*j!*x^j)^m) for m = 2, n >= 0 and k = 0..m*n+1.

A291448 Triangle read by rows, denominators of coefficients (in rising powers) of rational polynomials P(n,x) such that Integral_{x=0..1} P'(n,x) = BernoulliMedian(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 5, 1, 1, 1, 3, 1, 5, 1, 7, 1, 1, 1, 3, 1, 5, 1, 7, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 11, 1, 1, 1, 3, 1, 5, 1, 7, 1, 1, 1, 11, 1, 13, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 1, 11, 1, 13, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 1, 11, 1, 13, 1, 1

Offset: 0

Peter Luschny, Aug 24 2017

See A291447 and A290694 for comments.

			Triangle starts:
[1, 1]
[1, 1, 1, 3]
[1, 1, 1, 3, 1, 5]
[1, 1, 1, 3, 1, 5, 1, 7]
[1, 1, 1, 3, 1, 5, 1, 7, 1, 1]
[1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 11]
[1, 1, 1, 3, 1, 5, 1, 7, 1, 1, 1, 11, 1, 13]

Cf. A164555/A027642, A212196/A181131, A291449/A291450, A290694/A290695, A291447/A291448.

Maple
```
# See A291447.
```

T[n_] := Integrate[Sum[(-1)^(n-j+1) StirlingS2[n, j] j! x^j, {j,0,n}]^2, x];
Trow[n_] := CoefficientList[T[n], x] // Denominator;
Table[Trow[r], {r, 0, 7}] // Flatten

T(n,k) = Denominator([x^k] Integral(Sum_{j=0..n}(-1)^(n-j)*Stirling2(n,j)*j!*x^j)^m) for m = 2, n >= 0 and k = 0..m*n+1.

A290694 Triangle read by rows, numerators of coefficients (in rising powers) of rational polynomials P(n, x) such that Integral_{x=0..1} P'(n, x) = Bernoulli(n, 1).

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 0, -1, 2, 0, 0, 1, -2, 3, 0, 0, -1, 14, -9, 24, 0, 0, 1, -10, 75, -48, 20, 0, 0, -1, 62, -135, 312, -300, 720, 0, 0, 1, -42, 903, -1680, 2800, -2160, 630, 0, 0, -1, 254, -1449, 40824, -21000, 27360, -17640, 4480

Offset: 0

Peter Luschny, Aug 24 2017

Consider a family of integrals I_m(n) = Integral_{x=0..1} P'(n, x)^m with P'(n,x) = Sum_{k=0..n}(-1)^(n-k)*Stirling2(n, k)*k!*x^k (see A278075 for the coefficients).
I_1(n) are the Bernoulli numbers A164555/A027642, I_2(n) are the Bernoulli median numbers A212196/A181131, I_3(n) are the numbers A291449/A291450.
The coefficients of the polynomials P_n(x)^m are for m = 1 A290694/A290695 and for m = 2 A291447/A291448.
Only omega(Clausen(n)) = A001221(A160014(n,1)) = A067513(n) coefficients are rational numbers if n is even. For odd n > 1 there are two rational coefficients.
Let C_k(n) = [x^k] P_n(x), k > 0 and n even. Conjecture: k is a prime factor of Clausen(n) <=> k = denominator(C_k(n)) <=> k does not divide Stirling2(n, k-1)*(k-1)!. (Note that by a comment in A019538 Stirling2(n, k-1)*(k-1)! is the number of chain topologies on an n-set having k open sets.)

			Triangle starts:
[0, 1]
[0, 0,  1]
[0, 0, -1,   2]
[0, 0,  1,  -2,    3]
[0, 0, -1,  14,   -9,  24]
[0, 0,  1, -10,   75, -48,   20]
[0, 0, -1,  62, -135, 312, -300, 720]
The first few polynomials are:
P_0(x) = x.
P_1(x) =  (1/2)*x^2.
P_2(x) = -(1/2)*x^2 + (2/3)*x^3.
P_3(x) =  (1/2)*x^2 - 2*x^3 + (3/2)*x^4.
P_4(x) = -(1/2)*x^2 + (14/3)*x^3 - 9*x^4 + (24/5)*x^5.
P_5(x) =  (1/2)*x^2 - 10*x^3 + (75/2)*x^4 - 48*x^5 + 20*x^6.
P_6(x) = -(1/2)*x^2 + (62/3)*x^3 - 135*x^4 + 312*x^5 - 300*x^6 + (720/7)*x^7.
Evaluated at x = 1 this gives an additive decomposition of the Bernoulli numbers:
B(0) =     1 =    1.
B(1) =   1/2 =  1/2.
B(2) =   1/6 = -1/2 +  2/3.
B(3) =     0 =  1/2 -    2 + 3/2.
B(4) = -1/30 = -1/2 + 14/3 -    9 + 24/5.
B(5) =     0 =  1/2 -   10 + 75/2 -   48 +  20.
B(6) =  1/42 = -1/2 + 62/3 -  135 +  312 - 300 + 720/7.

Peter Luschny, Illustrating A290694.

Cf. A164555/A027642, A212196/A181131, A291449/A291450, A290694/A290695, A291447/A291448.

Cf. A160014, A278075.

BG_row := proc(m, n, frac, val) local F, g, v;
F := (n, x) -> add((-1)^(n-k)*Stirling2(n,k)*k!*x^k, k=0..n):
g := x -> int(F(n,x)^m, x):
`if`(val = "val", subs(x=1, g(x)), [seq(coeff(g(x),x,j), j=0..m*n+1)]):
`if`(frac = "num", numer(%), denom(%)) end:
seq(BG_row(1, n, "num", "val"), n=0..16);         # A164555
seq(BG_row(1, n, "den", "val"), n=0..16);         # A027642
seq(print(BG_row(1, n, "num", "poly")), n=0..12); # A290694 (this seq.)
seq(print(BG_row(1, n, "den", "poly")), n=0..12); # A290695
# Alternatively:
T_row := n -> numer(PolynomialTools:-CoefficientList(add((-1)^(n-j+1)*Stirling2(n, j-1)*(j-1)!*x^j/j, j=1..n+1), x)): for n from 0 to 6 do T_row(n) od;

T[n_, k_] := If[k > 0, Numerator[StirlingS2[n, k - 1]*(k - 1)! / k], 0]; Table[T[n, k], {n, 0, 8}, {k, 0, n+1}] // Flatten

T(n, k) = Numerator(Stirling2(n, k - 1)*(k - 1)!/k) if k > 0 else 0; for n >= 0 and 0 <= k <= n+1.

A290695 Triangle read by rows, denominators of coefficients (in rising powers) of rational polynomials P(n, x) such that Integral_{x=0..1} P'(n, x) = Bernoulli(n, 1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 5, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 7, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 5, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Peter Luschny, Aug 24 2017

See A290694 for comments.

			Triangle starts:
[1, 1]
[1, 1, 2]
[1, 1, 2, 3]
[1, 1, 2, 1, 2]
[1, 1, 2, 3, 1, 5]
[1, 1, 2, 1, 2, 1, 1]
[1, 1, 2, 3, 1, 1, 1, 7]
[1, 1, 2, 1, 2, 1, 1, 1, 1]

Cf. A164555/A027642, A212196/A181131, A291449/A291450, A290694/A290695, A291447/A291448.

T_row := n -> denom(PolynomialTools:-CoefficientList(add((-1)^(n-j+1)*Stirling2(n, j-1)*(j-1)!*x^j/j, j=1..n+1), x)): for n from 0 to 7 do T_row(n) od;

T[n_] := Denominator[CoefficientList[Sum[(-1)^(n-j+1) StirlingS2[n, j-1] (j-1)! x^j/j, {j, 1, n+1}], x]];
Table[T[n], {n, 0, 7}] (* Jean-François Alcover, Jun 15 2019, from Maple *)

T(n, k) = Denominator([x^k] Integral (Sum_{j=0..n} (-1)^(n-j)*Stirling2(n,j)*j!* x^j)^m) for m = 1 and k = 0..n+1.

A291449 Numerators of Integral_{x=0..1} P(n, x)^3 with P(n, x) = Sum_{k=0..n} (-1)^(n-k)* Stirling2(n, k)k!x^k.

Original entry on oeis.org

1, 1, 13, 1, 43, -61, 728877, 81739, -1779449713, -2112052153, 730622680308569, 113221320488699, -3660430816956396309, -3021604582205161, 21842539561810574341396283, 66747470298418575790593659, -124586733960451680357554181608419, -28471605423890788373026535240299

Offset: 0

Peter Luschny, Aug 24 2017

Consider a family of integrals I(m, n) = Integral_{x=0..1} P(n, x)^m with P(n, x) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*k!*x^k. I(1, n) are the Bernoulli numbers A164555/A027642, I(2, n) are the Bernoulli median numbers A212196/A181131, I(3, n) are the numbers A291449/A291450. The coefficients of the polynomials P(n, x)^m are for m = 1 A290694/A290695, for m = 2 A291447/A291448. (See A290694 for further comments.)

Cf. A164555/A027642, A212196/A181131, A291449/A291450, A290694/A290695, A291447/A291448.

# Function BG_row is defined in A290694.
seq(BG_row(3, n, "num", "val"), n=0..17);

P[n_, x_] := Sum[(-1)^(n-k)*StirlingS2[n, k]*k!*x^k, {k, 0, n}];
a[n_] := Integrate[P[n, x]^3, {x, 0, 1}] // Numerator;
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jun 15 2019 *)

A291450 Denominators of Integral_{x=0..1} P(n, x)^3 with P(n, x) = Sum_{k=0..n}(-1)^(n-k)* Stirling2(n, k)k!x^k.

Original entry on oeis.org

1, 4, 140, 28, 20020, 4004, 6466460, 184756, 148728580, 29745716, 133706993420, 2431036244, 449741705140, 31885268, 670910837521540, 134182167504308, 409926521725660940, 4822664961478364, 1278006214791766460, 1921813856829724, 242081282475556183660, 4401477863191930612

Offset: 0

Peter Luschny, Aug 24 2017

See A291449 and A290694 for comments.

Cf. A164555/A027642, A212196/A181131, A291449/A291450, A290694/A290695, A291447/A291448.

# Function BG_row is defined in A290694.
seq(BG_row(3, n, "den", "val"), n=0..20);

P[n_, x_] := Sum[(-1)^(n-k)*StirlingS2[n, k]*k!*x^k, {k, 0, n}];
a[n_] := Integrate[P[n, x]^3, {x, 0, 1}] // Denominator;
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jun 15 2019 *)

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]

cp's OEIS Frontend

A240776 Define a square array B(m,n) (m>=0, n>=0) by B(n, n) = A212196(n)/A181131(n), B(n, n+1) = -A212196(n)/A181131(n), B(m, n) = B(m, n-1) + B(m+1, n-1); a(n) = numerator of B(0,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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A291447 Triangle read by rows, numerators of coefficients (in rising powers) of rational polynomials P(n, x) such that Integral_{x=0..1} P'(n, x) = BernoulliMedian(n).

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A291448 Triangle read by rows, denominators of coefficients (in rising powers) of rational polynomials P(n,x) such that Integral_{x=0..1} P'(n,x) = BernoulliMedian(n).

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A290694 Triangle read by rows, numerators of coefficients (in rising powers) of rational polynomials P(n, x) such that Integral_{x=0..1} P'(n, x) = Bernoulli(n, 1).

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A290695 Triangle read by rows, denominators of coefficients (in rising powers) of rational polynomials P(n, x) such that Integral_{x=0..1} P'(n, x) = Bernoulli(n, 1).

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A291449 Numerators of Integral_{x=0..1} P(n, x)^3 with P(n, x) = Sum_{k=0..n} (-1)^(n-k)* Stirling2(n, k)k!x^k.

A291450 Denominators of Integral_{x=0..1} P(n, x)^3 with P(n, x) = Sum_{k=0..n}(-1)^(n-k)* Stirling2(n, k)k!x^k.