A048999 -id:A048999 - cp's OEIS frontend

A053383 Triangle T(n,k) giving denominator of coefficient of x^(n-k) in Bernoulli polynomial B(n, x), n >= 0, 0 <= k <= n.

1, 1, 2, 1, 1, 6, 1, 2, 2, 1, 1, 1, 1, 1, 30, 1, 2, 3, 1, 6, 1, 1, 1, 2, 1, 2, 1, 42, 1, 2, 2, 1, 6, 1, 6, 1, 1, 1, 3, 1, 3, 1, 3, 1, 30, 1, 2, 1, 1, 5, 1, 1, 1, 10, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 66, 1, 2, 6, 1, 1, 1, 1, 1, 2, 1, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2730, 1, 2, 1, 1, 6, 1, 7, 1, 10, 1, 3, 1, 210, 1

Offset: 0

N. J. A. Sloane, Jan 06 2000

			The polynomials B(0,x), B(1,x), B(2,x), ... are 1; x - 1/2; x^2 - x + 1/6; x^3 - (3/2)*x^2 + (1/2)*x; x^4 - 2*x^3 + x^2 - 1/30; x^5 - (5/2)*x^4 + (5/3)*x^3 - (1/6)*x; x^6 - 3*x^5 + (5/2)*x^4 - (1/2)*x^2 + 1/42; ...
Triangle A053382/A053383 begins:
  1;
  1, -1/2;
  1,  -1,  1/6;
  1, -3/2, 1/2, 0;
  1,  -2,   1,  0, -1/30;
  1, -5/2, 5/3, 0, -1/6, 0;
  1,  -3,  5/2, 0, -1/2, 0, 1/42;
  ...
Triangle A196838/A196839 begins (this is the reflected version):
    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;
  ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 809.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 48, [14a].
M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 53.
H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 19, equations 19:4:1 - 19:4:8 at page 169.

T. D. Noe, Rows n = 0..50 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Bakir Farhi, Formulas Involving Bernoulli and Stirling Numbers of Both Kinds, Journal of Integer Sequences, Vol. 28 (2025), Article 25.2.6. See p. 16.
D. H. Lehmer, A new approach to Bernoulli polynomials, The American mathematical monthly 95.10 (1988): 905-911.
Index entries for sequences related to Bernoulli numbers.

Three versions of coefficients of Bernoulli polynomials: A053382/A053383; for reflected version see A196838/A196839; see also A048998 and A048999.

Cf. A144845 (lcm of row n).

with(ListTools): with(PolynomialTools):
CoeffList := p -> Reverse(CoefficientList(p, x)):
Trow := n -> denom(CoeffList(bernoulli(n, x))):
Flatten([seq(Trow(n), n = 0..13)]); # Peter Luschny, Apr 10 2021

t[n_, k_] := Denominator[ Coefficient[ BernoulliB[n, x], x, n - k]]; Flatten[ Table[t[n, k], {n, 0, 13}, {k, 0, n}]] (* Jean-François Alcover, Jan 15 2013 *)

v=[];for(n=0,6,v=concat(v,apply(denominator,Vec(bernpol(n)))));v \\ Charles R Greathouse IV, Jun 08 2012

More terms from James Sellers, Jan 10 2000

A196838 Numerators of coefficients of Bernoulli polynomials with rising powers of the variable.

Original entry on oeis.org

1, -1, 1, 1, -1, 1, 0, 1, -3, 1, -1, 0, 1, -2, 1, 0, -1, 0, 5, -5, 1, 1, 0, -1, 0, 5, -3, 1, 0, 1, 0, -7, 0, 7, -7, 1, -1, 0, 2, 0, -7, 0, 14, -4, 1, 0, -3, 0, 2, 0, -21, 0, 6, -9, 1, 5, 0, -3, 0, 5, 0, -7, 0, 15, -5, 1, 0, 5, 0, -11, 0, 11, 0, -11, 0, 55, -11, 1

Offset: 0

Wolfdieter Lang, Oct 23 2011

The denominator triangle is found under A196839.
This is the row reversed triangle A053382.
From Wolfdieter Lang, Oct 25 2011: (Start)
This is the Sheffer triangle (z/(exp(z)-1),z), meaning that the column e.g.f.'s are as given below in the formula section. In Roman's book `The Umbral Calculus`, Ch. 2, 5., p. 26ff this is called Appell for (exp(t)-1)/t (see A048854 for the reference).
The e.g.f. for the a- and z-sequence for this Sheffer triangle is 1 and (x-exp(x)+1)/x^2, respectively. See the link under A006232 for the definition. The z-sequence is z(n) = -1/(2*A000217(n+1)). This leads to the recurrence relations given below.
The e.g.f. for the row sums is x/(1-exp(-x)), leading to the rational sequence A164555(n)/A027664(n). The e.g.f. of the alternating row sums is
  x/(exp(x)*(exp(x)-1)), leading to the rational sequence
  (-1)^n*A164558(n)/A027664(n).
(End)

			The triangle starts with
n\m 0  1  2  3  4  5  6  7  8 ...
0:  1
1: -1  1
2:  1 -1  1
3:  0  1 -3  1
4: -1  0  1 -2  1
5:  0 -1  0  5 -5  1
6:  1  0 -1  0  5 -3  1
7:  0  1  0 -7  0  7 -7  1
8: -1  0  2  0 -7  0 14 -4  1
...
The rational triangle a(n,m)/A196839(n,m) starts with:
n\m   0     1     2    3    4    5     6    7   8 ...
0:    1
1:  -1/2    1
2:   1/6   -1     1
3:    0    1/2  -3/2   1
4:  -1/30   0     1   -2    1
5:    0   -1/6    0   5/3 -5/2   1
6:   1/42   0   -1/2   0   5/2  -3     1
7:    0    1/6    0  -7/6   0   7/2  -7/2   1
8:  -1/30   0    2/3   0  -7/3   0   14/3  -4   1
...
E.g., Bernoulli(2,x) = (1/6)*x^0 - 1*x^1 + 1*x^2.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1991 (Seventh printing).Second ed. 1994.

Naho Kawasaki and Yasuo Ohno, The triangle algorithm for Bernoulli polynomials, Integers, vol. 23 (2023). (See figure 4.)
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers, arXiv:1707.04451 [math.NT], 2017.
D. H. Lehmer, A new approach to Bernoulli polynomials, The American mathematical monthly 95.10 (1988): 905-911.

Three versions of coefficients of Bernoulli polynomials: A053382/A053383; for reflected version see A196838/A196839; see also A048998 and A048999.

# Without using Maple's Bernoulli polynomials (Kawasaki and Ohno call it
# the 'triangle algorithm for B(n, x)'):
b := proc(n, m, x) option remember; if n = 0 then 1/(m + 1) else
normal((m + 1)*b(n-1, m + 1, x) - (m + 1 - x)*b(n-1, m, x)) fi end:
Bcoeffs := n -> local k; [seq(coeff(b(n, 0, x), x, k), k = 0..n)]:
for n from 0 to 8 do numer(Bcoeffs(n)) od; # Peter Luschny, Jun 16 2023

row[n_] := CoefficientList[BernoulliB[n, x], x] // Numerator;
Table[row[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jun 15 2018 *)

row(n) = apply(x->numerator(x), Vecrev(bernpol(n)));
tabl(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Jun 15 2018

T(n,m) = numerator([x^m]Bernoulli(n,x)), n>=0, m=0..n.
E.g.f. of Bernoulli(n,x): z*exp(x*z)/(exp(z)-1).
See the Graham et al. reference, eq. (7.80), p. 354.
From Wolfdieter Lang, Oct 25 2011: (Start)
The e.g.f. for column no. m>=0 of the rational triangle B(n,m):=a(n,m)/A096839(n,m) is x^(m+1)/(m!*(exp(x)-1)).
(see the Sheffer-Appell comment above).
The Sheffer a-sequence, given as comment above, leads to the recurrence r(n,m)=(n/m)*r(n-1,m-1), n>=1, m>=1. E.g., -1/6 = B(5,1) = (5/1)*B(4,0)= -5/30 = -1/6.
The Sheffer z-sequence, given as comment above, leads to the recurrence
B(n,0) = n*sum(z(j)*B(n-1,j),j=0..n-1), n>=1. B(0,0)=1.
E.g., -1/30 = B(4,0) = 4*((-1/2)*0 + (-1/6)*(1/2) + (-1/12)*(-3/2) + (-1/20)*1) = -1/30.
(End)
T(n,m) = numerator(binomial(n,m)*Bernoulli(n-m)). - Fabián Pereyra, Mar 04 2020

A196839 Triangle of denominators of the coefficient of x^m in the n-th Bernoulli polynomial, 0 <= m <= n.

Original entry on oeis.org

1, 2, 1, 6, 1, 1, 1, 2, 2, 1, 30, 1, 1, 1, 1, 1, 6, 1, 3, 2, 1, 42, 1, 2, 1, 2, 1, 1, 1, 6, 1, 6, 1, 2, 2, 1, 30, 1, 3, 1, 3, 1, 3, 1, 1, 1, 10, 1, 1, 1, 5, 1, 1, 2, 1, 66, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 1, 2, 1, 1, 1, 1, 1, 6, 2, 1, 2730, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1

Offset: 0

Wolfdieter Lang, Oct 23 2011

The numerator triangle is found under A196838.

This is the row reversed triangle A053383.

			The triangle starts with
n\m 0  1  2  3  4  5  6  7  8 ...
0:  1
1:  2  1
2:  6  1  1
3:  1  2  2  1
4: 30  1  1  1  1
5:  1  6  1  3  2  1
6: 42  1  2  1  2  1  1
7:  1  6  1  6  1  2  2  1
8: 30  1  3  1  3  1  3  1  1
...
For the start of the rational triangle A196838(n,m)/a(n,m) see the example section in A196838.

D. H. Lehmer, A new approach to Bernoulli polynomials, The American mathematical monthly 95.10 (1988): 905-911.

Three versions of coefficients of Bernoulli polynomials: A053382/A053383; for reflected version see A196838/A196839; see also A048998 and A048999.

with(ListTools):with(PolynomialTools):
CoeffList := p -> CoefficientList(p, x):
Trow := n -> denom(CoeffList(bernoulli(n, x))):
Flatten([seq(Trow(n), n = 0..12)]); # Peter Luschny, Apr 10 2021

T(n,m) = denominator([x^m]Bernoulli(n,x)), n>=0, m=0..n.
E.g.f. of Bernoulli(n,x): z*exp(x*z)/(exp(z)-1).
See the Graham et al. reference given in A196838, eq. (7.80), p. 354.
T(n,m) = denominator(binomial(n,m)*Bernoulli(n-m)). - Fabián Pereyra, Mar 04 2020

Name edited by M. F. Hasler, Mar 09 2020

A053382 Triangle T(n,k) giving numerator of coefficient of x^(n-k) in Bernoulli polynomial B(n, x), n >= 0, 0<=k<=n.

Original entry on oeis.org

1, 1, -1, 1, -1, 1, 1, -3, 1, 0, 1, -2, 1, 0, -1, 1, -5, 5, 0, -1, 0, 1, -3, 5, 0, -1, 0, 1, 1, -7, 7, 0, -7, 0, 1, 0, 1, -4, 14, 0, -7, 0, 2, 0, -1, 1, -9, 6, 0, -21, 0, 2, 0, -3, 0, 1, -5, 15, 0, -7, 0, 5, 0, -3, 0, 5, 1, -11, 55, 0, -11, 0, 11, 0, -11, 0, 5, 0, 1, -6, 11, 0, -33, 0, 22, 0

Offset: 0

N. J. A. Sloane, Jan 06 2000

			The polynomials B(0,x), B(1,x), B(2,x), ... are 1; x-1/2; x^2-x+1/6; x^3-3/2*x^2+1/2*x; x^4-2*x^3+x^2-1/30; x^5-5/2*x^4+5/3*x^3-1/6*x; x^6-3*x^5+5/2*x^4-1/2*x^2+1/42; ...
Triangle A053382/A053383 begins:
  1,
  1, -1/2,
  1, -1, 1/6,
  1, -3/2, 1/2, 0,
  1, -2, 1, 0, -1/30,
  1, -5/2, 5/3, 0, -1/6, 0,
  1, -3, 5/2, 0, -1/2, 0, 1/42,
  ...
Triangle A196838/A196839 begins (this is the reflected version):
  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,
  ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 809.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 48, [14a].
H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 19, equations 19:4:1 - 19:4:8 at page 169.

T. D. Noe, Rows n=0..50 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Bakir Farhi, Formulas Involving Bernoulli and Stirling Numbers of Both Kinds, Journal of Integer Sequences, Vol. 28 (2025), Article 25.2.6. See p. 16.
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers, arXiv:1707.04451 [math.NT], 2017.
D. H. Lehmer, A new approach to Bernoulli polynomials, The American mathematical monthly 95.10 (1988): 905-911.
H. Pan and Z. W. Sun, New identities involving Bernoulli and Euler polynomials, arXiv:math/0407363 [math.NT], 2004.
Index entries for sequences related to Bernoulli numbers.

Three versions of coefficients of Bernoulli polynomials: A053382/A053383; for reflected version see A196838/A196839; see also A048998 and A048999.

Maple
```
with(numtheory); bernoulli(n,x);
```

t[n_, k_] := Numerator[ Coefficient[ BernoulliB[n, x], x, n-k]]; Flatten[ Table[t[n, k], {n, 0, 12}, {k, 0, n}]] (* Jean-François Alcover, Aug 07 2012 *)

v=[];for(n=0,6,v=concat(v,apply(numerator,Vec(bernpol(n)))));v \\ Charles R Greathouse IV, Jun 08 2012

B(m, x) = Sum_{n=0..m} 1/(n+1)*Sum_{k=0..n} (-1)^k*C(n, k)*(x+k)^m.

More terms from James Sellers, Jan 10 2000

A048998 Triangle giving coefficients of (n+1)!*B_n(x), where B_n(x) is a Bernoulli polynomial. Rising powers of x.

Original entry on oeis.org

1, -1, 2, 1, -6, 6, 0, 12, -36, 24, -4, 0, 120, -240, 120, 0, -120, 0, 1200, -1800, 720, 120, 0, -2520, 0, 12600, -15120, 5040, 0, 6720, 0, -47040, 0, 141120, -141120, 40320, -12096, 0, 241920, 0, -846720, 0, 1693440, -1451520, 362880

Offset: 0

N. J. A. Sloane

See A074909 for generators for the Bernoulli polynomials and connections to the beheaded Pascal triangle and reciprocals of the integers. - Tom Copeland, Nov 17 2014

			B_0(x)=1; B_1(x)=x-1/2; B_2(x)=x^2-x+1/6; B_3(x)=x^3-3*x^2/2+x/2; B_4(x)=x^4-2*x^3+x^2-1/30; ...
Triangle starts:
   1;
  -1,  2;
   1, -6,   6;
   0, 12, -36, 24;
  ...

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 5th ed., Section 9.62.

Cf. A048999, A074909.

A048998 := proc(n,k) coeftayl(bernoulli(n,x),x=0,k) ; (n+1)!*% ; end proc:
seq(seq(A048998(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Jun 27 2011
# second program:
b := proc(n, m, x) option remember; if n = 0 then 1/(m + 1) else
(n + 1) * ((m + 1)*b(n - 1, m + 1, x) - (m + 1 - x)*b(n - 1, m, x)) fi end:
row := n -> seq(coeff(b(n, 0, x), x, k), k = 0..n):
seq(row(n), n = 0..8); # Peter Luschny, Jun 20 2023

Flatten[Table[CoefficientList[(n + 1)! BernoulliB[n, x], x], {n, 0, 10}]] (* T. D. Noe, Jun 21 2011 *)

t*exp(x*t)/(exp(t)-1) = Sum_{n >= 0} B_n(x)*t^n/n!.

a(n,m) = [x^m]((n+1)!*B_n(x)), n>=0, m=0,...,n. - Wolfdieter Lang, Jun 21 2011

Added 'Rising powers of x' in name - Wolfdieter Lang, Jun 21 2011

A129378 Row sums of coefficients of Bernoulli twin number polynomials.

Original entry on oeis.org

1, 1, 4, 20, 116, 744, 5160, 39360, 350784, 3749760, 42940800, 442713600, 4650877440, 109244298240, 2833294464000, -3487131648000, -2166903606067200, 51809012320665600, 6808619561103360000, -131306587205713920000, -26982365129174827008000, 595860034297401409536000

Offset: 0

Paul Curtz, Jun 08 2007

sign

The origin of the sequence are polynomials on pages 61 and 69 of the CCSA paper. The first few of the polynomials have been noted in the 1992 Gazette paper.
We construct Bernoulli twin numbers polynomials C(n,x) = Sum_{j=1..n} binomial(n-1,j-1)*B(j,x) where B(n,x) are the Bernoulli polynomials of A048998 and A048999 and where binomial(.,.) is the Pascal triangle A007318: C(0,x)=B(0,x); C(1,x)=B(1,x); C(2,x)=B(2,x)+B(1,x); C(3,x)=B(3,x)+2B(2,x)+B(1,x).
The triangle of coefficients [x^m] C(n,x) for rows n=0,1,2,.. and decreasing power m=n,...,0 along each row starts
  1;
  1, -1/2;
  1,    0, -1/3;
  1,  1/2, -1/2, -1/6;
The rightmost fraction in row n, that is, the absolute term C(n,0), is the Bernoulli twin number C(n) of A129826(n), i.e., C(n) = A129826(n)/(n+1)!.
If rows are multiplied by (n+1)!, the triangle becomes
    1;
    2,  -1;
    6,   0,  -2;
   24,  12, -12,  -4;
  120, 120, -60, -60, -4;
The sequence a(n) gives the row sums of this triangle. The sums of antidiagonals are 1, 2, 5, 24, 130, 828, 6056.... The first column of the inverse of the triangle is 1, 2, 3, 3, 0, (0 continued).

P. Curtz, Integration numerique ..., Note no. 12 CCSA (later CELAR), 1969. (See A129841, A129696.)
P. Curtz, Gazette des Mathematiciens, 1992, no. 52, p. 44.

G. C. Greubel, Table of n, a(n) for n = 0..300

Cf. A000142, A007318, A048998, A048999, A129724, A129826.

f:= func< n | n le 2 select (-1)^Floor((n+1)/2)/(n+1) else (-1)^n*BernoulliNumber(Floor(n - (1-(-1)^n)/2)) >;
A129378:= func< n | n eq 0 select 1 else Factorial(n+1)*(f(n)+1) >;
[A129378(n): n in [0..30]]; // G. C. Greubel, Feb 01 2024

c[n_?EvenQ] := BernoulliB[n]; c[n_?OddQ] := -BernoulliB[n-1]; c[1] = -1/2; c[2] = -1/3; a[n_] := (n+1)!*(1+c[n]); a[0]=1; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Aug 08 2012, after given formula *)

def f(n): return (-1)^((n+1)//2)/(n+1) if n<3 else (-1)^n*bernoulli(n-(n%2))
def A129378(n): return 1 if n==0 else factorial(n+1)*(f(n)+1)
[A129378(n) for n in range(31)] # G. C. Greubel, Feb 01 2024

a(n) = (n+1)!*(1 + C(n)) = A129826(n) + A000142(n+1), n>0.

Edited and extended by R. J. Mathar, Aug 06 2008

cp's OEIS Frontend

A053383 Triangle T(n,k) giving denominator of coefficient of x^(n-k) in Bernoulli polynomial B(n, x), n >= 0, 0 <= k <= n.

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A196838 Numerators of coefficients of Bernoulli polynomials with rising powers of the variable.

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A196839 Triangle of denominators of the coefficient of x^m in the n-th Bernoulli polynomial, 0 <= m <= n.

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A053382 Triangle T(n,k) giving numerator of coefficient of x^(n-k) in Bernoulli polynomial B(n, x), n >= 0, 0<=k<=n.

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A048998 Triangle giving coefficients of (n+1)!*B_n(x), where B_n(x) is a Bernoulli polynomial. Rising powers of x.

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Maple

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A129378 Row sums of coefficients of Bernoulli twin number polynomials.

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