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
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
Programs
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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
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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
Comments