A053383 Triangle T(n,k) giving denominator of coefficient of x^(n-k) in Bernoulli polynomial B(n, x), n >= 0, 0 <= k <= n.

1, 1, 2, 1, 1, 6, 1, 2, 2, 1, 1, 1, 1, 1, 30, 1, 2, 3, 1, 6, 1, 1, 1, 2, 1, 2, 1, 42, 1, 2, 2, 1, 6, 1, 6, 1, 1, 1, 3, 1, 3, 1, 3, 1, 30, 1, 2, 1, 1, 5, 1, 1, 1, 10, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 66, 1, 2, 6, 1, 1, 1, 1, 1, 2, 1, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2730, 1, 2, 1, 1, 6, 1, 7, 1, 10, 1, 3, 1, 210, 1

Offset: 0

Author

N. J. A. Sloane, Jan 06 2000

Keywords

• nonn
• easy
• nice
• frac
• tabl

Examples

			The polynomials B(0,x), B(1,x), B(2,x), ... are 1; x - 1/2; x^2 - x + 1/6; x^3 - (3/2)*x^2 + (1/2)*x; x^4 - 2*x^3 + x^2 - 1/30; x^5 - (5/2)*x^4 + (5/3)*x^3 - (1/6)*x; x^6 - 3*x^5 + (5/2)*x^4 - (1/2)*x^2 + 1/42; ...
Triangle A053382/A053383 begins:
  1;
  1, -1/2;
  1,  -1,  1/6;
  1, -3/2, 1/2, 0;
  1,  -2,   1,  0, -1/30;
  1, -5/2, 5/3, 0, -1/6, 0;
  1,  -3,  5/2, 0, -1/2, 0, 1/42;
  ...
Triangle A196838/A196839 begins (this is the reflected version):
    1;
  -1/2,   1;
   1/6,  -1,    1;
    0,   1/2, -3/2,  1;
  -1/30,  0,    1,  -2,    1;
    0,  -1/6,   0,  5/3, -5/2,  1;
   1/42,  0,  -1/2,  0,   5/2, -3, 1;
  ...
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 809.
• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 48, [14a].
• M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 53.
• H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.
• Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 19, equations 19:4:1 - 19:4:8 at page 169.

Links

• T. D. Noe, Rows n = 0..50 of triangle, flattened
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• Bakir Farhi, Formulas Involving Bernoulli and Stirling Numbers of Both Kinds, Journal of Integer Sequences, Vol. 28 (2025), Article 25.2.6. See p. 16.
• D. H. Lehmer, A new approach to Bernoulli polynomials, The American mathematical monthly 95.10 (1988): 905-911.
• Index entries for sequences related to Bernoulli numbers.

Crossrefs

Three versions of coefficients of Bernoulli polynomials: A053382/A053383; for reflected version see A196838/A196839; see also A048998 and A048999.

Cf. A144845 (lcm of row n).

Programs

Maple

with(ListTools): with(PolynomialTools):
CoeffList := p -> Reverse(CoefficientList(p, x)):
Trow := n -> denom(CoeffList(bernoulli(n, x))):
Flatten([seq(Trow(n), n = 0..13)]); # Peter Luschny, Apr 10 2021

Mathematica

t[n_, k_] := Denominator[ Coefficient[ BernoulliB[n, x], x, n - k]]; Flatten[ Table[t[n, k], {n, 0, 13}, {k, 0, n}]] (* Jean-François Alcover, Jan 15 2013 *)

PARI

v=[];for(n=0,6,v=concat(v,apply(denominator,Vec(bernpol(n)))));v \\ Charles R Greathouse IV, Jun 08 2012

Extensions

More terms from James Sellers, Jan 10 2000