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%I #57 Jun 16 2023 17:53:11
%S 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,
%T 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,
%U 5,0,-7,0,15,-5,1,0,5,0,-11,0,11,0,-11,0,55,-11,1
%N Numerators of coefficients of Bernoulli polynomials with rising powers of the variable.
%C The denominator triangle is found under A196839.
%C This is the row reversed triangle A053382.
%C From _Wolfdieter Lang_, Oct 25 2011: (Start)
%C 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).
%C 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.
%C 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
%C x/(exp(x)*(exp(x)-1)), leading to the rational sequence
%C (-1)^n*A164558(n)/A027664(n).
%C (End)
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1991 (Seventh printing).Second ed. 1994.
%H Naho Kawasaki and Yasuo Ohno, <a href="http://math.colgate.edu/~integers/x39/x39.pdf">The triangle algorithm for Bernoulli polynomials</a>, Integers, vol. 23 (2023). (See figure 4.)
%H Wolfdieter Lang, <a href="https://arxiv.org/abs/1707.04451">On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers</a>, arXiv:1707.04451 [math.NT], 2017.
%H D. H. Lehmer, <a href="http://www.jstor.org/stable/2322383">A new approach to Bernoulli polynomials</a>, The American mathematical monthly 95.10 (1988): 905-911.
%F T(n,m) = numerator([x^m]Bernoulli(n,x)), n>=0, m=0..n.
%F E.g.f. of Bernoulli(n,x): z*exp(x*z)/(exp(z)-1).
%F See the Graham et al. reference, eq. (7.80), p. 354.
%F From _Wolfdieter Lang_, Oct 25 2011: (Start)
%F 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)).
%F (see the Sheffer-Appell comment above).
%F 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.
%F The Sheffer z-sequence, given as comment above, leads to the recurrence
%F B(n,0) = n*sum(z(j)*B(n-1,j),j=0..n-1), n>=1. B(0,0)=1.
%F E.g., -1/30 = B(4,0) = 4*((-1/2)*0 + (-1/6)*(1/2) + (-1/12)*(-3/2) + (-1/20)*1) = -1/30.
%F (End)
%F T(n,m) = numerator(binomial(n,m)*Bernoulli(n-m)). - _Fabián Pereyra_, Mar 04 2020
%e The triangle starts with
%e n\m 0 1 2 3 4 5 6 7 8 ...
%e 0: 1
%e 1: -1 1
%e 2: 1 -1 1
%e 3: 0 1 -3 1
%e 4: -1 0 1 -2 1
%e 5: 0 -1 0 5 -5 1
%e 6: 1 0 -1 0 5 -3 1
%e 7: 0 1 0 -7 0 7 -7 1
%e 8: -1 0 2 0 -7 0 14 -4 1
%e ...
%e The rational triangle a(n,m)/A196839(n,m) starts with:
%e n\m 0 1 2 3 4 5 6 7 8 ...
%e 0: 1
%e 1: -1/2 1
%e 2: 1/6 -1 1
%e 3: 0 1/2 -3/2 1
%e 4: -1/30 0 1 -2 1
%e 5: 0 -1/6 0 5/3 -5/2 1
%e 6: 1/42 0 -1/2 0 5/2 -3 1
%e 7: 0 1/6 0 -7/6 0 7/2 -7/2 1
%e 8: -1/30 0 2/3 0 -7/3 0 14/3 -4 1
%e ...
%e E.g., Bernoulli(2,x) = (1/6)*x^0 - 1*x^1 + 1*x^2.
%p # Without using Maple's Bernoulli polynomials (Kawasaki and Ohno call it
%p # the 'triangle algorithm for B(n, x)'):
%p b := proc(n, m, x) option remember; if n = 0 then 1/(m + 1) else
%p normal((m + 1)*b(n-1, m + 1, x) - (m + 1 - x)*b(n-1, m, x)) fi end:
%p Bcoeffs := n -> local k; [seq(coeff(b(n, 0, x), x, k), k = 0..n)]:
%p for n from 0 to 8 do numer(Bcoeffs(n)) od; # _Peter Luschny_, Jun 16 2023
%t row[n_] := CoefficientList[BernoulliB[n, x], x] // Numerator;
%t Table[row[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Jun 15 2018 *)
%o (PARI) row(n) = apply(x->numerator(x), Vecrev(bernpol(n)));
%o tabl(nn) = for (n=0, nn, print(row(n))); \\ _Michel Marcus_, Jun 15 2018
%Y Three versions of coefficients of Bernoulli polynomials: A053382/A053383; for reflected version see A196838/A196839; see also A048998 and A048999.
%K sign,easy,tabl,frac
%O 0,9
%A _Wolfdieter Lang_, Oct 23 2011
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