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 A053383 Triangle T(n,k) giving denominator of coefficient of x^(n-k) in Bernoulli polynomial B(n, x), n >= 0, 0<=k<=n. 14

%I

%S 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,

%T 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,

%U 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

%N Triangle T(n,k) giving denominator 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 M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 53.

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

%H T. D. Noe, <a href="/A053383/b053383.txt">Rows n=0..50 of triangle, flattened</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

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

%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_] := 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 *)

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

%Y Cf. A144845 (lcm of row n)

%K nonn,easy,nice,frac,tabl

%O 0,3

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

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

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