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

0,3

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.

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].

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.

Example

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;
  ...

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 *)

Prog

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

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).

Sequence in context: A096162 A333144 A306297 * A181538 A322128 A125731

Adjacent sequences: A053380 A053381 A053382 * A053384 A053385 A053386

Keyword

nonn,easy,nice,frac,tabl

Author

N. J. A. Sloane, Jan 06 2000

Extensions

More terms from James A. Sellers, Jan 10 2000

Status

approved