cp's OEIS Frontend

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

%I A053382 #60 Jul 02 2025 16:01:59
%S A053382 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,
%T A053382 -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,
%U A053382 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
%N A053382 Triangle T(n,k) giving numerator of coefficient of x^(n-k) in Bernoulli polynomial B(n, x), n >= 0, 0<=k<=n.
%D A053382 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.
%D A053382 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 48, [14a].
%D A053382 H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.
%D A053382 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.
%H A053382 T. D. Noe, <a href="/A053382/b053382.txt">Rows n=0..50 of triangle, flattened</a>
%H A053382 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A053382 Bakir Farhi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Farhi/farhi26.html">Formulas Involving Bernoulli and Stirling Numbers of Both Kinds</a>, Journal of Integer Sequences, Vol. 28 (2025), Article 25.2.6. See p. 16.
%H A053382 Wolfdieter Lang, <a href="https://arxiv.org/abs/1707.04451">On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers</a>, arXiv:1707.04451 [math.NT], 2017.
%H A053382 D. H. Lehmer, <a href="http://www.jstor.org/stable/2322383">A new approach to Bernoulli polynomials</a>, The American mathematical monthly 95.10 (1988): 905-911.
%H A053382 H. Pan and Z. W. Sun, <a href="http://arXiv.org/abs/math.NT/0407363">New identities involving Bernoulli and Euler polynomials</a>, arXiv:math/0407363 [math.NT], 2004.
%H A053382 <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers</a>.
%F A053382 B(m, x) = Sum_{n=0..m} 1/(n+1)*Sum_{k=0..n} (-1)^k*C(n, k)*(x+k)^m.
%e A053382 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; ...
%e A053382 Triangle A053382/A053383 begins:
%e A053382   1,
%e A053382   1, -1/2,
%e A053382   1, -1, 1/6,
%e A053382   1, -3/2, 1/2, 0,
%e A053382   1, -2, 1, 0, -1/30,
%e A053382   1, -5/2, 5/3, 0, -1/6, 0,
%e A053382   1, -3, 5/2, 0, -1/2, 0, 1/42,
%e A053382   ...
%e A053382 Triangle A196838/A196839 begins (this is the reflected version):
%e A053382   1,
%e A053382   -1/2, 1,
%e A053382   1/6, -1, 1,
%e A053382   0, 1/2, -3/2, 1,
%e A053382   -1/30, 0, 1, -2, 1,
%e A053382   0, -1/6, 0, 5/3, -5/2, 1,
%e A053382   1/42, 0, -1/2, 0, 5/2, -3, 1,
%e A053382   ...
%p A053382 with(numtheory); bernoulli(n,x);
%t A053382 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 *)
%o A053382 (PARI) v=[];for(n=0,6,v=concat(v,apply(numerator,Vec(bernpol(n)))));v \\ _Charles R Greathouse IV_, Jun 08 2012
%Y A053382 Three versions of coefficients of Bernoulli polynomials: A053382/A053383; for reflected version see A196838/A196839; see also A048998 and A048999.
%K A053382 sign,easy,nice,frac,tabl
%O A053382 0,8
%A A053382 _N. J. A. Sloane_, Jan 06 2000
%E A053382 More terms from _James Sellers_, Jan 10 2000