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

0,8

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.

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].

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.

Formula

B(m, x) = Sum{n=0..m, 1/(n+1)*Sum[k=0..n, (-1)^k*C(n, k)*(x+k)^m ]].

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(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 *)

Prog

(PARI) v=[]; for(n=0, 6, v=concat(v, apply(numerator, 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.

Sequence in context: A183134 A328747 A346061 * A031253 A291624 A291635

Adjacent sequences: A053379 A053380 A053381 * A053383 A053384 A053385

Keyword

sign,easy,nice,frac,tabl

Author

N. J. A. Sloane, Jan 06 2000

Extensions

More terms from James A. Sellers, Jan 10 2000

Status

approved