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A053382
Triangle T(n,k) giving numerator of coefficient of x^(n-k) in Bernoulli polynomial B(n, x), n >= 0, 0<=k<=n.
21
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.
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
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.
H. Pan and Z. W. Sun, New identities involving Bernoulli and Euler polynomials, arXiv:math/0407363 [math.NT], 2004.
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 * A380822 A031253 A378929
KEYWORD
sign,easy,nice,frac,tabl
AUTHOR
N. J. A. Sloane, Jan 06 2000
EXTENSIONS
More terms from James Sellers, Jan 10 2000
STATUS
approved