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%I #74 Feb 06 2021 05:17:33
%S 1,0,-1,0,-1,3,0,0,1,-1,0,2,5,-30,15,0,0,-2,-5,10,-3,0,-16,-42,91,315,
%T -315,63,0,0,16,42,-7,-105,63,-9,0,144,404,-540,-2345,-840,3150,-1260,
%U 135,0,0,-144,-404,-100,665,448,-630,180,-15,0,-768,-2288,2068,11792,8195,-8085,-8778,6930,-1485,99
%N Triangle read by rows giving coefficients in Bernoulli polynomials as defined in A001898, after multiplication by the common denominators A001898(n).
%C Let p(n, x) = Sum_{k=0..n} T(n, k)*x^k, then the polynomials (-1)^n*p(n; x)/x are called 'Stirling polynomials' by Knuth et al. (CMath, eq. 6.45). - _Peter Luschny_, Feb 05 2021
%D Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math., 1st ed.; Addison-Wesley, 1989, p. 257.
%H F. N. David, <a href="https://archive.org/details/probabilitytheor033214mbp/page/n113">Probability Theory for Statistical Methods</a>, Cambridge, 1949; see pp. 103-104. [There is an error in the recurrence for B_s^{(r)}.]
%H N. E. Nörlund, <a href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN373206070">Vorlesungen ueber Differenzenrechnung</a> Springer 1924, (p. 146).
%F E.g.f.: (y/(exp(y)-1))^x. - _Vladeta Jovovic_, Feb 27 2006
%F Let p(n, x) = (Sum_{k=0..n} E2(n, k)*binomial(x + k, 2*n))/(Product_{j=1..n} (j-x)), where E2 are the second-order Eulerian numbers (A201637), then T(n, k) = [x^k] M(n+1)*p(n, x), where M(n) are the Minkowski numbers (A053657). - _Peter Luschny_, Feb 05 2021
%e The Bernoulli polynomials B(0)(x) through B(6)(x) are:
%e 1
%e -(1/2)* x
%e (1/12)*(3*x - 1)*x
%e -(1/8)*(x-1)*x^2
%e (1/240)*(15*x^3 - 30*x^2 + 5*x + 2)*x
%e -(1/96)*(x-1)*(3*x^2 - 7*x - 2)*x^2
%e (1/4032)*(63*x^5 - 315*x^4 + 315*x^3 + 91*x^2 - 42*x - 16)*x
%e Triangle of coefficients starts:
%e [0] [1],
%e [1] [0, -1],
%e [2] [0, -1, 3],
%e [3] [0, 0, 1, -1],
%e [4] [0, 2, 5, -30, 15],
%e [5] [0, 0, -2, -5, 10, -3],
%e [6] [0, -16, -42, 91, 315, -315, 63],
%e [7] [0, 0, 16, 42, -7, -105, 63, -9],
%e [8] [0, 144, 404, -540, -2345, -840, 3150, -1260, 135].
%p CoeffList := p -> op(PolynomialTools:-CoefficientList(simplify(p),x)):
%p E2 := (n, k) -> combinat[eulerian2](n, k): m := n -> mul(j-x, j = 1..n):
%p Epoly := n -> simplify(expand(add(E2(n, k)*binomial(x+k,2*n), k = 0..n)/m(n))):
%p poly := n -> Epoly(n)*denom(Epoly(n)):
%p seq(print(CoeffList(poly(n))), n = 0..8); # _Peter Luschny_, Feb 05 2021
%t row[n_] := NorlundB[n, x] // Together // Numerator // CoefficientList[#, x]&; Table[row[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Jul 06 2019, after _Peter Luschny_ *)
%o (Sage) # Formula (83), page 146 in Nörlund.
%o @cached_function
%o def NoerlundB(n, x):
%o if n == 0: return 1
%o return expand((-x/n)*add((-1)^k*binomial(n,k)*bernoulli(k)*NoerlundB(n-k,x) for k in (1..n)))
%o def A100655_row(n): return numerator(NoerlundB(n, x)).list()
%o [A100655_row(n) for n in (0..8)] # _Peter Luschny_, Jul 01 2019
%Y Cf. A001898, A027641, A027642, A053657, A100615, A100616, A201637.
%K sign,tabl
%O 0,6
%A _N. J. A. Sloane_, Dec 05 2004
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