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%I #41 Jun 24 2017 00:28:29
%S 1,2,-1,6,-6,1,24,-36,12,0,120,-240,120,0,-4,720,-1800,1200,0,-120,0,
%T 5040,-15120,12600,0,-2520,0,120,40320,-141120,141120,0,-47040,0,6720,
%U 0,362880,-1451520,1693440,0,-846720,0,241920,0,-12096,3628800
%N Triangle giving coefficients of (n+1)!*B_n(x), where B_n(x) is a Bernoulli polynomial, ordered by falling powers of x.
%D I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 5th ed., Section 9.62.
%H T. D. Noe, <a href="/A048999/b048999.txt">Rows n = 0..50, flattened</a>
%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>
%F t*exp(x*t)/(exp(t)-1) = Sum_{n >= 0} B_n(x)*t^n/n!.
%F a(n,m) = [x^(n-m)]((n+1)!*B_n(x)), n>=0, m=0,...,n. - _Wolfdieter Lang_, Jun 21 2011
%e B_0=1 => a(0) = 1;
%e B_1(x)=x-1/2 => a(1..2) = 2, -1;
%e B_2(x)=x^2-x+1/6 => a(3..5) = 6, -6, 1;
%e B_3(x)=x^3-3*x^2/2+x/2 => a(6..9) = 24, -36, 12, 0;
%e B_4(x)=x^4-2*x^3+x^2-1/30 => a(10..14) = 120, -240, 120, 0, -4;
%e ...
%t row[n_] := (n+1)!*Reverse[ CoefficientList[ BernoulliB[n, x], x]]; Flatten[ Table[ row[n], {n, 0, 9}]] (* _Jean-François Alcover_, Feb 17 2012 *)
%o (PARI) P=Pol(t*exp(x*t)/(exp(t)-1)); for(i=0,15, z=polcoeff(P,i,t)*i!; print(z" => ",(i+1)!*Vec(z))) /* print B_n's and list of normalized coefficients */ \\ _M. F. Hasler_, Jun 21 2011
%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,tabl
%O 0,2
%A _N. J. A. Sloane_
%E Name clarified by adding 'Falling powers of x.' from _Wolfdieter Lang_, Jun 21 2011
%E Values corrected by inserting a(9),a(20),a(35)=0 by _M. F. Hasler_, Jun 21 2011
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