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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. 20

%I

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

%D 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 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 48, [14a].

%D H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.

%H T. D. Noe, <a href="/A053382/b053382.txt">Rows n=0..50 of triangle, flattened</a>

%H 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 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 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 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 <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>

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

%e 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 Triangle A053382/A053383 begins:

%e 1,

%e 1, -1/2,

%e 1, -1, 1/6,

%e 1, -3/2, 1/2, 0,

%e 1, -2, 1, 0, -1/30,

%e 1, -5/2, 5/3, 0, -1/6, 0,

%e 1, -3, 5/2, 0, -1/2, 0, 1/42,

%e ...

%e Triangle A196838/A196839 begins (this is the reflected version):

%e 1,

%e -1/2, 1,

%e 1/6, -1, 1,

%e 0, 1/2, -3/2, 1,

%e -1/30, 0, 1, -2, 1,

%e 0, -1/6, 0, 5/3, -5/2, 1,

%e 1/42, 0, -1/2, 0, 5/2, -3, 1,

%e ...

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

%t t[n_, k_] := Numerator[ Coefficient[ BernoulliB[n, x], x, n-k]]; Flatten[ Table[t[n, k], {n, 0, 12}, {k, 0, n}]] (* _Jean-Fran├žois Alcover_, Aug 07 2012 *)

%o (PARI) v=[];for(n=0,6,v=concat(v,apply(numerator,Vec(bernpol(n)))));v \\ _Charles R Greathouse IV_, Jun 08 2012

%Y Three versions of coefficients of Bernoulli polynomials: A053382/A053383; for reflected version see A196838/A196839; see also A048998 and A048999.

%K sign,easy,nice,frac,tabl

%O 0,8

%A _N. J. A. Sloane_, Jan 06 2000

%E More terms from _James A. Sellers_, Jan 10 2000

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Last modified June 4 08:18 EDT 2020. Contains 334825 sequences. (Running on oeis4.)