A100655 Triangle read by rows giving coefficients in Bernoulli polynomials as defined in A001898, after multiplication by the common denominators A001898(n).

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, -315, 63, 0, 0, 16, 42, -7, -105, 63, -9, 0, 144, 404, -540, -2345, -840, 3150, -1260, 135, 0, 0, -144, -404, -100, 665, 448, -630, 180, -15, 0, -768, -2288, 2068, 11792, 8195, -8085, -8778, 6930, -1485, 99

Offset: 0

Author

N. J. A. Sloane, Dec 05 2004

Keywords

• sign
• tabl

Comments

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

Examples

			The Bernoulli polynomials B(0)(x) through B(6)(x) are:
        1
    -(1/2)* x
    (1/12)*(3*x - 1)*x
    -(1/8)*(x-1)*x^2
   (1/240)*(15*x^3 - 30*x^2 + 5*x + 2)*x
   -(1/96)*(x-1)*(3*x^2 - 7*x - 2)*x^2
  (1/4032)*(63*x^5 - 315*x^4 + 315*x^3 + 91*x^2 - 42*x - 16)*x
Triangle of coefficients starts:
[0] [1],
[1] [0,  -1],
[2] [0,  -1,   3],
[3] [0,   0,   1,   -1],
[4] [0,   2,   5,  -30,    15],
[5] [0,   0,  -2,   -5,    10,   -3],
[6] [0, -16, -42,   91,   315, -315,   63],
[7] [0,   0,  16,   42,    -7, -105,   63,    -9],
[8] [0, 144, 404, -540, -2345, -840, 3150, -1260, 135].
		

References

• Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math., 1st ed.; Addison-Wesley, 1989, p. 257.

Links

• F. N. David, Probability Theory for Statistical Methods, Cambridge, 1949; see pp. 103-104. [There is an error in the recurrence for B_s^{(r)}.]
• N. E. Nörlund, Vorlesungen ueber Differenzenrechnung Springer 1924, (p. 146).

Crossrefs

Cf. A001898, A027641, A027642, A053657, A100615, A100616, A201637.

Programs

Maple

CoeffList := p -> op(PolynomialTools:-CoefficientList(simplify(p),x)):
E2 := (n, k) -> combinat[eulerian2](n, k): m := n -> mul(j-x, j = 1..n):
Epoly := n -> simplify(expand(add(E2(n, k)*binomial(x+k,2*n), k = 0..n)/m(n))):
poly := n -> Epoly(n)*denom(Epoly(n)):
seq(print(CoeffList(poly(n))), n = 0..8); # Peter Luschny, Feb 05 2021

Mathematica

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 *)

Sage

# Formula (83), page 146 in Nörlund.
@cached_function
def NoerlundB(n, x):
    if n == 0: return 1
    return expand((-x/n)*add((-1)^k*binomial(n,k)*bernoulli(k)*NoerlundB(n-k,x) for k in (1..n)))
def A100655_row(n): return numerator(NoerlundB(n, x)).list()
[A100655_row(n) for n in (0..8)] # Peter Luschny, Jul 01 2019

Formula

E.g.f.: (y/(exp(y)-1))^x. - Vladeta Jovovic, Feb 27 2006

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