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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.

%I A053383 #50 Jul 02 2025 16:01:59
%S A053383 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 A053383 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 A053383 1,2,6,1,1,1,1,1,2,1,6,1,1,1,1,1,2,1,1,1,2,1,1,1,2730,1,2,1,1,6,1,7,1,10,1,3,1,210,1
%N A053383 Triangle T(n,k) giving denominator of coefficient of x^(n-k) in Bernoulli polynomial B(n, x), n >= 0, 0 <= k <= n.
%D A053383 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 A053383 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 48, [14a].
%D A053383 M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 53.
%D A053383 H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.
%D A053383 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 A053383 T. D. Noe, <a href="/A053383/b053383.txt">Rows n = 0..50 of triangle, flattened</a>
%H A053383 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 A053383 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 A053383 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 A053383 <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers</a>.
%e A053383 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 A053383 Triangle A053382/A053383 begins:
%e A053383   1;
%e A053383   1, -1/2;
%e A053383   1,  -1,  1/6;
%e A053383   1, -3/2, 1/2, 0;
%e A053383   1,  -2,   1,  0, -1/30;
%e A053383   1, -5/2, 5/3, 0, -1/6, 0;
%e A053383   1,  -3,  5/2, 0, -1/2, 0, 1/42;
%e A053383   ...
%e A053383 Triangle A196838/A196839 begins (this is the reflected version):
%e A053383     1;
%e A053383   -1/2,   1;
%e A053383    1/6,  -1,    1;
%e A053383     0,   1/2, -3/2,  1;
%e A053383   -1/30,  0,    1,  -2,    1;
%e A053383     0,  -1/6,   0,  5/3, -5/2,  1;
%e A053383    1/42,  0,  -1/2,  0,   5/2, -3, 1;
%e A053383   ...
%p A053383 with(ListTools): with(PolynomialTools):
%p A053383 CoeffList := p -> Reverse(CoefficientList(p, x)):
%p A053383 Trow := n -> denom(CoeffList(bernoulli(n, x))):
%p A053383 Flatten([seq(Trow(n), n = 0..13)]); # _Peter Luschny_, Apr 10 2021
%t A053383 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 A053383 (PARI) v=[];for(n=0,6,v=concat(v,apply(denominator,Vec(bernpol(n)))));v \\ _Charles R Greathouse IV_, Jun 08 2012
%Y A053383 Three versions of coefficients of Bernoulli polynomials: A053382/A053383; for reflected version see A196838/A196839; see also A048998 and A048999.
%Y A053383 Cf. A144845 (lcm of row n).
%K A053383 nonn,easy,nice,frac,tabl
%O A053383 0,3
%A A053383 _N. J. A. Sloane_, Jan 06 2000
%E A053383 More terms from _James Sellers_, Jan 10 2000