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

0,9

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)

References

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

Links

Table of n, a(n) for n=0..77.

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.

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

Example

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.

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 *)

Prog

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

Crossrefs

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

Sequence in context: A228488 A338638 A062172 * A284376 A088205 A318923

Adjacent sequences: A196835 A196836 A196837 * A196839 A196840 A196841

Keyword

sign,easy,tabl,frac

Author

Wolfdieter Lang, Oct 23 2011

Status

approved