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

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

Author

Wolfdieter Lang, Oct 23 2011

Keywords

• sign
• easy
• tabl
• frac

Comments

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)

Examples

			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.
		

References

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

Links

• 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.

Crossrefs

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

Programs

Maple

# 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

Mathematica

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

PARI

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

Formula

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