A053382 Triangle T(n,k) giving numerator of coefficient of x^(n-k) in Bernoulli polynomial B(n, x), n >= 0, 0<=k<=n.

1, 1, -1, 1, -1, 1, 1, -3, 1, 0, 1, -2, 1, 0, -1, 1, -5, 5, 0, -1, 0, 1, -3, 5, 0, -1, 0, 1, 1, -7, 7, 0, -7, 0, 1, 0, 1, -4, 14, 0, -7, 0, 2, 0, -1, 1, -9, 6, 0, -21, 0, 2, 0, -3, 0, 1, -5, 15, 0, -7, 0, 5, 0, -3, 0, 5, 1, -11, 55, 0, -11, 0, 11, 0, -11, 0, 5, 0, 1, -6, 11, 0, -33, 0, 22, 0

Offset: 0

Author

N. J. A. Sloane, Jan 06 2000

Keywords

• sign
• 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].
• 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.
• 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.
• H. Pan and Z. W. Sun, New identities involving Bernoulli and Euler polynomials, arXiv:math/0407363 [math.NT], 2004.
• 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.

Programs

Maple

with(numtheory); bernoulli(n,x);

Mathematica

t[n_, k_] := Numerator[ Coefficient[ BernoulliB[n, x], x, n-k]]; Flatten[ Table[t[n, k], {n, 0, 12}, {k, 0, n}]] (* Jean-François Alcover, Aug 07 2012 *)

PARI

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

Formula

B(m, x) = Sum_{n=0..m} 1/(n+1)*Sum_{k=0..n} (-1)^k*C(n, k)*(x+k)^m.

Extensions

More terms from James Sellers, Jan 10 2000