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
(End)
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.
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
KEYWORD
AUTHOR
Wolfdieter Lang, Oct 23 2011
STATUS
approved