A053383 -id:A053383 - cp's OEIS frontend

A162299 Faulhaber's triangle: triangle T(k,y) read by rows, giving denominator of the coefficient [m^y] of the polynomial Sum_{x=1..m} x^(k-1).

1, 2, 2, 6, 2, 3, 1, 4, 2, 4, 30, 1, 3, 2, 5, 1, 12, 1, 12, 2, 6, 42, 1, 6, 1, 2, 2, 7, 1, 12, 1, 24, 1, 12, 2, 8, 30, 1, 9, 1, 15, 1, 3, 2, 9, 1, 20, 1, 2, 1, 10, 1, 4, 2, 10, 66, 1, 2, 1, 1, 1, 1, 1, 6, 2, 11, 1, 12, 1, 8, 1, 6, 1, 8, 1, 12, 2, 12, 2730, 1, 3, 1, 10, 1, 7, 1, 6, 1, 1, 2, 13, 1, 420, 1, 12, 1, 20, 1, 28, 1, 60, 1, 12, 2, 14, 6, 1, 90, 1, 6, 1, 10, 1, 18, 1, 30, 1, 6, 2, 15

Offset: 0

Juri-Stepan Gerasimov, Jun 30 2009 and Jul 02 2009

There are many versions of Faulhaber's triangle: search the OEIS for his name. For example, A220862/A220963 is essentially the same as this triangle, except for an initial column of 0's. - N. J. A. Sloane, Jan 28 2017

			The first few polynomials:
    m;
   m/2  + m^2/2;
   m/6  + m^2/2 + m^3/3;
    0   + m^2/4 + m^3/2 + m^4/4;
  -m/30 +   0   + m^3/3 + m^4/2 + m^5/5;
  ...
Initial rows of Faulhaber's triangle of fractions H(n, k) (n >= 0, 1 <= k <= n+1):
    1;
   1/2,  1/2;
   1/6,  1/2,  1/3;
    0,   1/4,  1/2,  1/4;
  -1/30,  0,   1/3,  1/2,  1/5;
    0,  -1/12,  0,   5/12, 1/2,  1/6;
   1/42,  0,  -1/6,   0,   1/2,  1/2,  1/7;
    0,   1/12,  0,  -7/24,  0,   7/12, 1/2,  1/8;
  -1/30,  0,   2/9,   0,  -7/15,  0,   2/3,  1/2,  1/9;
  ...
The triangle starts in row k=1 with columns 1<=y<=k as
     1
     2   2
     6   2  3
     1   4  2  4
    30   1  3  2  5
     1  12  1 12  2  6
    42   1  6  1  2  2  7
     1  12  1 24  1 12  2  8
    30   1  9  1 15  1  3  2  9
     1  20  1  2  1 10  1  4  2 10
    66   1  2  1  1  1  1  1  6  2 11
     1  12  1  8  1  6  1  8  1 12  2 12
  2730   1  3  1 10  1  7  1  6  1  1  2 13
     1 420  1 12  1 20  1 28  1 60  1 12  2 14
     6   1 90  1  6  1 10  1 18  1 30  1  6  2 15
  ...
Initial rows of triangle of fractions:
    1;
   1/2, 1/2;
   1/6, 1/2,  1/3;
    0,  1/4,  1/2,  1/4;
  -1/30, 0,   1/3,  1/2,  1/5;
    0, -1/12,  0,   5/12, 1/2,  1/6;
   1/42, 0,  -1/6,   0,   1/2,  1/2,  1/7;
    0,  1/12,  0,  -7/24,  0,   7/12, 1/2,  1/8;
  -1/30, 0,   2/9,   0,  -7/15,  0,   2/3,  1/2,  1/9;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Mohammad Torabi-Dashti, Faulhaber’s Triangle, College Math. J., 42:2 (2011), 96-97.
Mohammad Torabi-Dashti, Faulhaber’s Triangle [Annotated scanned copy of preprint]
Eric Weisstein's MathWorld, Power Sum

Cf. A000367, A162298 (numerators).
See also A220962/A220963.
Cf. A053382, A053383.

A162299 := proc(k,y) local gf,x; gf := sum(x^(k-1),x=1..m) ; coeftayl(gf,m=0,y) ; denom(%) ; end proc: # R. J. Mathar, Jan 24 2011
# To produce Faulhaber's triangle of fractions H(n,k) (n >= 0, 1 <= k <= n+1):
H:=proc(n,k) option remember; local i;
if n<0 or k>n+1 then 0;
elif n=0 then 1;
elif k>1 then (n/k)*H(n-1,k-1);
else 1 - add(H(n,i),i=2..n+1); fi; end;
for n from 0 to 10 do lprint([seq(H(n,k),k=1..n+1)]); od:
for n from 0 to 12 do lprint([seq(numer(H(n,k)),k=1..n+1)]); od: # A162298
for n from 0 to 12 do lprint([seq(denom(H(n,k)),k=1..n+1)]); od: # A162299 # N. J. A. Sloane, Jan 28 2017

H[n_, k_] := H[n, k] = Which[n < 0 || k > n+1, 0, n == 0, 1, k > 1, (n/k)* H[n - 1, k - 1], True, 1 - Sum[H[n, i], {i, 2, n + 1}]];
Table[H[n, k] // Denominator, {n, 0, 14}, {k, 1, n + 1}] // Flatten (* Jean-François Alcover, Aug 04 2022 *)

Faulhaber's triangle of fractions H(n,k) (n >= 0, 1 <= k <= n+1) is defined by: H(0,1)=1; for 2<=k<=n+1, H(n,k) = (n/k)*H(n-1,k-1) with H(n,1) = 1 - Sum_{i=2..n+1}H(n,i). - N. J. A. Sloane, Jan 28 2017

Sum_{x=1..m} x^(k-1) = (Bernoulli(k,m+1)-Bernoulli(k))/k.

Offset set to 0 by Alois P. Heinz, Feb 19 2021

A285068 Denominators of the generalized Bernoulli numbers B[3,1] = 3^n*B(n, 1/3).

Original entry on oeis.org

1, 2, 2, 1, 10, 1, 14, 1, 10, 1, 22, 1, 910, 1, 2, 1, 170, 1, 266, 1, 110, 1, 46, 1, 910, 1, 2, 1, 290, 1, 4774, 1, 170, 1, 2, 1, 639730, 1, 2, 1, 4510, 1, 602, 1, 230, 1, 94, 1, 15470, 1, 22

Offset: 0

Wolfdieter Lang, Apr 28 2017

The numerators are given in A157799.
Because B(n, 2/3) = (-1)^n*B(n, 1/3) (from the e.g.f. z*exp(x*z)/(exp(z)-1) of Bernoulli polynomials {B(n, x)}_{n>=0}) one has for the numbers B[3,2](n) = 3^n*B(n, 2/3) the numerators (-1)^n*A157799(n) and the denominators a(n).
This sequence gives also the denominators of {3^n*B(n)}_{n>=0} with numerators given in A285863.

			The Bernoulli numbers r(n) = B[3,1](n) begin: 1, -1/2, -1/2, 1, 13/10, -5, -121/14, 49, 1093/10, -809, -49205/22, 20317, 61203943/910, -722813, -5580127/2, 34607305, ...
The Bernoulli numbers B[3,2](n) begin: 1, 1/2, -1/2, -1, 13/10, 5, -121/14, -49, 1093/10, 809, -49205/22, -20317, 61203943/910, 722813, -5580127/2, -34607305, ...
From _Peter Luschny_, Mar 26 2021: (Start)
The generalized Bernoulli numbers as given in the Luschny link are different.
1, -7/2, 23/2, -35, 973/10, -245, 7943/14, -1295, 31813/10, -7721, 288715/22, -13475, 128296423/910, -882557, -4891999/2, 33870025, ...
The numerators of these numbers are in A157811. (End)

Indranil Ghosh, Table of n, a(n) for n = 0..500
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers, arXiv:1707.04451 [math.NT], 2017.
Peter Luschny, Generalized Bernoulli numbers.

Cf. A053382/A053383, A157799, A157800, A196838/A196839, A282629, A284861, A285863.

Cf. A144845, A157811.

Table[Denominator[3^n*BernoulliB[n, 1/3]], {n, 0, 100}] (* Indranil Ghosh, Jul 18 2017 *)

a(n) = denominator(3^n * bernfrac(n)); \\ Ruud H.G. van Tol, Jan 31 2024

from sympy import bernoulli, Rational
def a(n):
    return (3**n * bernoulli(n, Rational(1,3))).as_numer_denom()[1]
print([a(n) for n in range(101)])  # Indranil Ghosh, Jul 18 2017

# uses [gen_bernoulli_number from A157811]
print([denominator((-1)^n*gen_bernoulli_number(n, 3)) for n in range(23)])
# Peter Luschny, Mar 26 2021

a(n) = denominator(r(n)) with the rationals (in lowest terms) r(n) = Sum_{k=0..n} ((-1)^k / (k+1))*A282629(n, k)*k! = Sum_{k=0..n} ((-1)^k/(k+1))*A284861(n, k). r(n) = B[3,1](n) = 3^n*B(n, 1/3) with the Bernoulli polynomials A196838/A196839 or A053382/A053383.

a(n) = A157800(n)/3^n, n >= 0.

A157817 Numerator of Bernoulli(n, 1/4).

Original entry on oeis.org

1, -1, -1, 3, 7, -25, -31, 427, 127, -12465, -2555, 555731, 1414477, -35135945, -57337, 2990414715, 118518239, -329655706465, -5749691557, 45692713833379, 91546277357, -7777794952988025, -1792042792463, 1595024111042171723, 1982765468311237, -387863354088927172625

Offset: 0

N. J. A. Sloane, Nov 08 2009

From Wolfdieter Lang, Apr 28 2017: (Start)
The rationals r(n) = Sum_{k=0..n} ((-1)^k / (k+1))*A285061(n, k)*k! = Sum_{k=0..n} ((-1)^k/(k+1))*A225473(n, k) define generalized Bernoulli numbers, named B[4,1](n), in terms of the generalized Stirling2 numbers S2[4,1]. The numerators of r(n) are a(n) and the denominators A141459(n). r(n) = B[4,1](n) = 4^n*B(n, 1/4) with the Bernoulli polynomials B(n, x) = Bernoulli(n, x) from A196838/A196839 or A053382/A053383.
The generalized Bernoulli numbers B[4,3](n) = Sum_{k=0..n} ((-1)^k/(k+1))* A225467(n, k)*k! = Sum_{k=0..n} ((-1)^k/(k+1))*A225473(n, k) satisfy
  B[4,3](n) = 4^n*B(n, 3/4) = (-1)^n*B[4,1](n). They have numerators (-1)^n*a(n) and also denominators A141459(n). (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..250

For denominators see A157818 and A141459.

Table[Numerator[BernoulliB[n, 1/4]], {n, 0, 50}] (* Vincenzo Librandi, Mar 16 2014 *)

From Wolfdieter Lang, Apr 28 2017: (Start)
a(n) = numerator(Bernoulli(n, 1/4)) with denominator A157818(n) (see the name).
a(n) = numerator(4^n*Bernoulli(n, 1/4)) with denominator A141459(n) = A157818(n)/4^n.
a(n)*(-1)^n = numerator(4^n*Bernoulli(n, 3/4)) with denominator A141459(n).
(End)

A157866 Numerator of Bernoulli(n, 1/5).

Original entry on oeis.org

1, -3, 1, 6, -29, -74, 4537, 1946, -23789, -88434, 15034541, 6154786, -10417027559, -607884394, 13199705071, 80834386026, -34108052679853, -13923204233954, 51709981061257363, 3015393801263666, -1029159167703800359, -801997872697905114, 629565265428734672873

Offset: 0

N. J. A. Sloane, Nov 08 2009

From Wolfdieter Lang, Jul 05 2017: (Start)
a(n) gives also the numerators of the generalized Bernoulli numbers B[5,1](n) = 5^n*B(n, 1/5) with the Bernoulli polynomials B(n, x) = Bernoulli(n, x) from A196838/A196839 or A053382/A053383. For the denominators see A288872(n) = A157867(n)/5^n.
(-1)^n*a(n) gives the numerators of the generalized Bernoulli numbers B[5,4](n). The denominators are also A288872(n).
The generalized Bernoulli numbers B[d,a](n), for d >= 1, a >= 0, with gcd(d, a) = 1 are defined in terms of generalized Stirling2 numbers by B[d,a](n) = Sum_{k=0..n} ((-1)^k / (k+1))*S2[d,a](n, k)*k!, n >= 0. See A285061 for more details.
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..250

For denominators see A157867, and also A288872.

Cf. A053382/A053383, A196838/A196839, A285061.

Table[Numerator[BernoulliB[n, 1/5]], {n, 0, 50}] (* Vincenzo Librandi, Mar 16 2014 *)

a(n)=numerator(subst(bernpol(n, x), x, 1/5)); \\ Michel Marcus, Jul 06 2017

A178252 Triangle T(n,m) read by rows: the numerator of the coefficient [x^m] of the umbral inverse Bernoulli polynomials B^{-1}(n,x), 0 <= m <= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 1, 5, 10, 5, 1, 1, 1, 3, 5, 5, 3, 1, 1, 1, 7, 7, 35, 7, 7, 1, 1, 1, 4, 28, 14, 14, 28, 4, 1, 1, 1, 9, 12, 21, 126, 21, 12, 9, 1, 1, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1, 1, 11, 55, 165, 66, 77, 66, 165, 55, 11, 1, 1, 1, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 1

Offset: 0

Paul Curtz, May 24 2010

The fractions A053382(n,m)/A053383(n,m) give the triangle of the coefficients of the Bernoulli polynomials:
    1;
  -1/2,   1;
   1/6,  -1,    1;
    0,   1/2, -3/2,  1;
  -1/30,  0,    1,  -2,    1;
    0,  -1/6,   0,  5/3, -5/2,  1;
   1/42,  0,  -1/2,  0,   5/2, -3, 1;
The matrix inverse of this triangle defines coefficients of the umbral inverse Bernoulli polynomials B^{-1}(n,x) in row n:
   1;
  1/2,  1;
  1/3,  1,  1;
  1/4,  1, 3/2,   1;
  1/5,  1,  2,    2,   1;
  1/6,  1, 5/2, 10/3, 5/2,   1;
  1/7,  1,  3,    5,   5,    3,   1;
  1/8,  1, 7/2,   7, 35/4,   7,  7/2,  1;
  1/9,  1,  4,  28/3, 14,   14, 28/3,  4,  1;
  1/10, 1, 9/2,  12,  21, 126/5, 21,  12, 9/2, 1;
  1/11, 1,  5,   15,  30,   42,  42,  30, 15,  5, 1;
The current triangle T(n,m) is the numerator of the entry in row n and column m.
In the majority of cases, T(n,m) = A050169(n,m), but since we use the numerators of the reduced fractions, an integer factor may be missing in this equation.
Umbral composition (e.g., B(.,x)^k = B(k,x)) gives B^(-1)(n,B(.,x)) = x^n = B(n,B^(-1)(.,x)). - Tom Copeland, Aug 25 2015

			The triangle T(n,k) begins:
n\k 0 1  2  3   4   5   6    7   8   9  10 11 12 13
0:  1
1:  1 1
2:  1 1  1
3:  1 1  3  1
4:  1 1  2  2   1
5:  1 1  5 10   5   1
6:  1 1  3  5   5   3   1
7:  1 1  7  7  35   7   7    1
8:  1 1  4 28  14  14  28    4   1
9:  1 1  9 12  21 126  21   12   9   1
10: 1 1  5 15  30  42  42   30  15   5   1
11: 1 1 11 55 165  66  77   66 165  55  11  1
12: 1 1  6 22  55  99 132  132  99  55  22  6  1
13: 1 1 13 26 143 143 429 1716 429 143 143 26 13  1
... reformatted. - _Wolfdieter Lang_, Aug 25 2015

Cf. A178340 (denominators).

Cf. A074909, A258820, A003989, A053382, A053383.

nm := 15 : eM := Matrix(nm,nm) :
for n from 0 to nm-1 do for m from 0 to n do eM[n+1,m+1] :=coeff(bernoulli(n,x),x,m) ; end do: for m from n+1 to nm-1 do eM[n+1,m+1] := 0 ; end do: end do:
eM := LinearAlgebra[MatrixInverse](eM) :
for n from 1 to nm do for m from 1 to n do printf("%a,", numer(eM[n,m])) ; end do: end do: # R. J. Mathar, Dec 21 2010

max = 13; coes = Table[ PadRight[ CoefficientList[ BernoulliB[n, x], x], max], {n, 0, max-1}]; inv = Inverse[coes]; Table[ Take[inv[[n]], n], {n, 1, max}] // Flatten // Numerator (* Jean-François Alcover, Aug 09 2012 *)

tabl(nn) = {for (n=0, nn, for (k=0, n, print1(numerator(binomial(n+1,k)/(n+1)), ", ");); print(););} \\ after Tom Copeland comment; Michel Marcus, Jul 25 2015

"Palindromic:" T(n,m+1) = T(n,n-m). T(n,0)=1.
From Tom Copeland, Jun 18 2015: (Start)
The umbral inverse Bernoulli polynomials are Binv(n,x) = [(1+x)^(n+1)-x^(n+1)]/(n+1) with the e.g.f. e^(t*x) * (e^t-1)/t. (See A074909 for more details.) Therefore, T(n,k) is the numerator of the reduced fraction C(n+1,k)/(n+1) for 0 <= k < (n+1).
The reversed rows are presented as the diagonals of A258820.
T(n,k) = A258820(2n-k,n-k) = A003989(n+1,n+1-k) * n! / [ k! (n+1-k)! ], where A003989(j,k) = gcd(j,k). (End)
From Wolfdieter Lang, Aug 26 2015: (Start)
The following refers to the rational triangle TBinv with entries T(n,k)/A178340(n, m), n >= m >= 0.
The inverse of the Bernoulli triangle TB(n, m) with entries A196838(n,m)/A196839(n,m), n >= m >= 0, is the Sheffer triangle (z/(exp(z)-1),z). Therefore, the inverse triangle TBinv is the Sheffer triangle ((exp(z)-1)/z, z). This means that the e.g.f. of the sequence of column m of TBinv ((exp(x)-1)/x)*x^m/m! for m >= 0.
The e.g.f. of the row polynomials of TBinv, called Binv(n, x) = Sum_{m=0..n} TBinv(n,m)*x^m, is gBinv(z,x) = ((exp(z)-1)/z)*exp(x*z) (of the so-called Appell type).
The e.g.f. of the row sums is gBinv(x,1).
The e.g.f. of the alternating row sums is gBinv(x,-1) = (1 - exp(-x))/x.
The e.g.f. of the a-sequence of this Sheffer triangle is 1, and the e.g.f. of the z-sequence is (exp(x) - x -1)/((exp(x) -1)*x). This is the sequence 1/2, -1/12, 0, 1/120, 0, -1/252, 0, 1/240, 0, -1/132, .... For a- and z-sequences of Sheffer triangles and the corresponding recurrences see A006232.
The convolution property of the row polynomials Binv(n, x) is Binv(n, x+y) = Sum_{k=0..n} binomial(n, k)*Binv(n-k, x)*y^n (or with x and y exchanged).
The row polynomials satisfy (d/dx)Binv(n, x) = n*Binv(n-1, x), with Binv(0, x) = 1 (from Meixner's identity).
(End)

Redefined based on reduced fractions by R. J. Mathar, Dec 21 2010

The term umbral was added by Tom Copeland, Aug 25 2015

A285863 Numerators of Bernoulli numbers 3^n*B(n), with B(n) = A027641(n)/A027642(n).

Original entry on oeis.org

1, -3, 3, 0, -27, 0, 243, 0, -2187, 0, 98415, 0, -122408577, 0, 11160261, 0, -51899996619, 0, 5664991530321, 0, -202943637014337, 0, 8938507796555139, 0, -22252066887294301257, 0, 7246946747292751629, 0, -181103830292539169071623

Offset: 0

Wolfdieter Lang, Apr 29 2017

The denominators are given in A285068.
In general the numbers B(d;n) = d^n*B(n), for n >= 0, have e.g.f. d*x/(exp(d*x) - 1). They are also the exponential convolution of the generalized Bernoulli numbers B[d,a](n), obtained from the generalized Stirling2 numbers S2[d,a], with the sequence {(-a)^n}_{n>=0}. See a comment in A157817 for the  B[4,1] and B[4,3] examples.
These numbers B(d;n) and their polynomials B(d;n,x) = Sum_{m=0..n} binomial(n, m)*B(d;n-m)*x^m are used in the generalized so-called Faulhaber formula for the sums of powers of arithmetic progressions defined by SP(d,a;n,m) := Sum_{j=0..m} (a + d*j)^n = Sum_{k=0..n} binomial(n, k)*a^(n-k)*d^k*SP(k,m) with SP(k,m) = SP(1,0;k,m), n >= 0, m >= 0, and 0^0 := 1.
The Faulhaber formula is: SP(d,a;n,m) = (1/(d*(n+1)))*[B(d;n+1,x = a+d*(m+1)) - B(d;n+1,x = d) - B(d;n+1,x = a)  + B(d;n+1,x=0) + d^(n+1)*[n=0]]. Here [n=0] is the Kronecker delta_{n,0} symbol: 1 if n=0 and 0 otherwise.
A simpler version of the Faulhaber formula is for a=0: SP(d,0;0,m) = m+1 and  SP(d,0;n,m) = d^n*(1/(n+1))*(B(n+1, x = m+1) - B(n+1, x=1)) for n >= 1, and for a an integer >= 1: Sum_{k=0..n} binomial(n, k)*a^(n-k) * d^k * (1/(k+1)) * (B(k+1, x=m+1) - B(k+1, x=1)). Here B(n, x) =  B(1;n,x) are the usual Bernoulli polynomials from A196838/A196839 or A053382/A053383.

Indranil Ghosh, Table of n, a(n) for n = 0..300
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers, arXiv:1707.04451 [math.NT], 2017.

Cf. A027641/A027642, A144845, A285068.

seq(numer(3^n*bernoulli(n)), n=0..28); # Peter Luschny, Jul 17 2017

Table[Numerator[3^n*BernoulliB[n]], {n, 0, 100}] (* Indranil Ghosh, Jul 18 2017 *)

a(n) = numerator(3^n * bernfrac(n)); \\ Ruud H.G. van Tol, Jan 31 2024

from sympy import bernoulli
def a(n): return -3 if n == 1 else (3**n * bernoulli(n)).numerator
print([a(n) for n in range(51)]) # Indranil Ghosh, Jul 18 2017

a(n) = numerator(r(n)) with r(n) = 3^n*A027641(n)/A027642(n), n >= 0.

E.g.f. {r(n)}_{n>=0}: 3*x/(exp(3*x) - 1).

A213615 Triangle read by rows, coefficients of the Bernoulli polynomials B_{n}(x) times A144845(n) in descending order of powers.

Original entry on oeis.org

1, 2, -1, 6, -6, 1, 2, -3, 1, 0, 30, -60, 30, 0, -1, 6, -15, 10, 0, -1, 0, 42, -126, 105, 0, -21, 0, 1, 6, -21, 21, 0, -7, 0, 1, 0, 30, -120, 140, 0, -70, 0, 20, 0, -1, 10, -45, 60, 0, -42, 0, 20, 0, -3, 0, 66, -330, 495, 0, -462, 0, 330, 0, -99, 0, 5, 6, -33

Offset: 0

Peter Luschny, Jun 16 2012

			b(0,x) =  1
b(1,x) =  2*x    -  1
b(2,x) =  6*x^2  -  6*x    + 1
b(3,x) =  2*x^3  -  3*x^2  + x
b(4,x) = 30*x^4  - 60*x^3  + 30*x^2  - 1
b(5,x) =  6*x^5  - 15*x^4  + 10*x^3  - x

T. D. Noe, Rows n = 0..100 of triangle, flattened
Peter Luschny, The Computation and Asymptotics of the Bernoulli numbers.

Cf. A053383, A144845, A213616.

seq(seq(coeff(denom(bernoulli(i,x))*bernoulli(i,x),x,i-j),j=0..i),i=0..12);

Flatten[Table[p = Reverse[CoefficientList[BernoulliB[n, x], x]]; (LCM @@ Denominator[p])*p, {n, 0, 10}]] (* T. D. Noe, Nov 07 2012 *)

T(n,k) = A144845(n)*[x^(n-k)]B{n}(x).

A349004 Decimal expansion of lim_{n->infinity} B(2n, n)/n^(2n), where B(n, x) is the n-th Bernoulli polynomial.

Original entry on oeis.org

3, 1, 3, 0, 3, 5, 2, 8, 5, 4, 9, 9, 3, 3, 1, 3, 0, 3, 6, 3, 6, 1, 6, 1, 2, 4, 6, 9, 3, 0, 8, 4, 7, 8, 3, 2, 9, 1, 2, 0, 1, 3, 9, 4, 1, 2, 4, 0, 4, 5, 2, 6, 5, 5, 5, 4, 3, 1, 5, 2, 9, 6, 7, 5, 6, 7, 0, 8, 4, 2, 7, 0, 4, 6, 1, 8, 7, 4, 3, 8, 2, 6, 7, 4, 6, 7, 9, 2, 4, 1, 4, 8, 0, 8, 5, 6, 3, 0, 2, 9, 4, 6, 7, 9, 4, 7

Offset: 0

Vaclav Kotesovec, Nov 05 2021

Asymptotic expansion: B(2*n,n) / n^(2*n) ~ c0 + c1/n + c2/n^2 + ..., where
c0 = A349004
c1 = -0.11332842437985451266688985513574347679739396134203607414578687657...
c2 = -0.02939332883129837328682967905833985820907100422772261310141242364...
In general, for k>=1, B(k*n,n) / n^(k*n) ~ k/(exp(k) - 1).

			0.313035285499331303636161246930847832912013941240452655543152967567084...

William Bell, Problem 4312, Crux Mathematicorum, Vol. 44, No. 2 (2018), pp. 69 and 71; Solution to Problem 4312, ibid., Vol. 45, No. 2 (2019), pp. 92-93.
Eric Weisstein's World of Mathematics, Bernoulli Polynomial.
Wikipedia, Bernoulli Polynomials.
Index entries for transcendental numbers.

Cf. A053382, A053383, A073747, A349003.

$MaxExtraPrecision = 1000; funs[n_] := BernoulliB[2 n, n]/n^(2 n); Do[Print[N[Sum[(-1)^(m + j)*funs[j*Floor[1000/m]] * j^(m - 1)/(j - 1)!/(m - j)!, {j, 1, m}], 110]], {m, 10, 100, 10}]
RealDigits[2/(E^2 - 1), 10, 110][[1]]

Equals 2/(exp(2)-1).
From Peter Luschny, Nov 05 2021: (Start)
Equals lim_{n->oo} (1/n) * Sum_{k=0..n-1} B(2*n, 1 + k/n) by J. L. Raabe's multiplication theorem.
Equals -2 * lim_{n->oo} HurwitzZeta(1 - 2*n, n) * n^(1 - 2*n). (End)
Equals A073747 - 1. - Alois P. Heinz, Nov 05 2021
Equals Sum_{k>=1} tanh(1/2^k)/2^k (Bell, 2018). - Amiram Eldar, Apr 12 2022

A157883 Numerator of Bernoulli(n, 2/5).

Original entry on oeis.org

1, -1, -11, 3, 91, -43, -12347, 1183, 62851, -54423, -39448591, 3799763, 27287144401, -375591203, -34562009741, 49954996743, 89299092717107, -8604866798383, -135379643536733633, 1863607913992123, 2694379428323830241, -495661415843787963, -1648224141847799919403

Offset: 0

N. J. A. Sloane, Nov 08 2009

From Wolfdieter Lang, Jul 05 2017: (Start)
a(n) gives also the numerators of the generalized Bernoulli numbers B[5,2](n) = 5^n*Bernoulli(n, 2/5) with the Bernoulli polynomials B(n, x) = Bernoulli(n, x) from A196838/A196839 or A053382/A053383. For the denominators see A288872(n) = A157867(n)/5^n. See a comment under A157866 for B[d,a](n).
(-1)^n*a(n) gives the numerators of the generalized Bernoulli numbers B[5,3](n); the denominators are A288872(n).
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..250

For denominators see A157867.

Cf. A288872.

Table[Numerator[BernoulliB[n, 2/5]], {n, 0, 50}] (* Vincenzo Librandi, Mar 16 2014 *)

a(n) = numerator(subst(bernpol(n, x), x, 2/5)); \\ Michel Marcus, Jul 06 2017

from sympy import bernoulli, Integer
def a(n): return bernoulli(n, Integer(2)/5).numerator
print([a(n) for n in range(51)]) # Indranil Ghosh, Jul 06 2017

A213616 Triangle read by rows, coefficients of the Bernoulli nabla polynomials BN_{n}(x) times A144845(n) in descending order of powers.

Original entry on oeis.org

1, 2, -3, 6, -18, 13, 2, -9, 13, -6, 30, -180, 390, -360, 119, 6, -45, 130, -180, 119, -30, 42, -378, 1365, -2520, 2499, -1260, 253, 6, -63, 273, -630, 833, -630, 253, -42, 30, -360, 1820, -5040, 8330, -8400, 5060, -1680, 239, 10, -135, 780, -2520, 4998, -6300

Offset: 0

Peter Luschny, Jun 16 2012

The polynomials BN_{n}(x) have the e.g.f. t*exp(t*(x-1))/(exp(t)-1). The adjunct 'nabla' in the name refers to the backward difference operation.
BN_{n}(1) are the Bernoulli numbers.
In the difference table of the Bernoulli polynomials the polynomials BN_{n}(x) appear as the top row (see the link).

			bn(0,x) =  1,
bn(1,x) =  2*x   -   3,
bn(2,x) =  6*x^2 -  18*x   +  13,
bn(3,x) =  2*x^3 -   9*x^2 +  13*x   -   6,
bn(4,x) = 30*x^4 - 180*x^3 + 390*x^2 - 360*x   + 119,
bn(5,x) =  6*x^5 -  45*x^4 + 130*x^3 - 180*x^2 + 119*x - 30.

Peter Luschny, The Computation and Asymptotics of the Bernoulli numbers.

A053383, A144845, A213615.

seq(seq(coeff(denom(bernoulli(i, x))*bernoulli(i, x - 1), x, i - j), j=0..i), i=0..12);

Table[If[i == 0, 1, 1/First[ FactorTerms[ BernoulliB[i, x]]]]*Table[ Coefficient[ Denominator[ BernoulliB[i, x]]*BernoulliB[i, x-1], x, i-j], {j, 0, i}], {i, 0, 12}] // Flatten (* Jean-François Alcover, Sep 27 2013, after Maple *)

T(n,k) = A144845(n)*[x^(n-k)]BN{n}(x).

cp's OEIS Frontend

A162299 Faulhaber's triangle: triangle T(k,y) read by rows, giving denominator of the coefficient [m^y] of the polynomial Sum_{x=1..m} x^(k-1).

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A285068 Denominators of the generalized Bernoulli numbers B[3,1] = 3^n*B(n, 1/3).

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A157817 Numerator of Bernoulli(n, 1/4).

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A157866 Numerator of Bernoulli(n, 1/5).

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A178252 Triangle T(n,m) read by rows: the numerator of the coefficient [x^m] of the umbral inverse Bernoulli polynomials B^{-1}(n,x), 0 <= m <= n.

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A285863 Numerators of Bernoulli numbers 3^n*B(n), with B(n) = A027641(n)/A027642(n).

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A213615 Triangle read by rows, coefficients of the Bernoulli polynomials B_{n}(x) times A144845(n) in descending order of powers.

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A349004 Decimal expansion of lim_{n->infinity} B(2n, n)/n^(2n), where B(n, x) is the n-th Bernoulli polynomial.