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%I #23 Aug 28 2019 17:50:44
%S 1,1,-1,1,-1,1,1,-1,1,-1,1,-5,17,-5,5,1,-5,41,-236,691,-691,1,-7,14,
%T -22,359,-7,7,1,-14,77,-293,1519,-1237,3617,-3617,1,-6,217,-1129,8487,
%U -6583,750167,-43867,43867,1,-5,23,-470,689,-28399,1540967,-1254146,174611,-174611,1,-55,209,-902,60511
%N Triangle of numerators of coefficients of Faulhaber polynomials used for sums of even powers.
%C The companion triangle with the denominators is A093559.
%C Sum_{k=1..n} k^(2*(m-1)) = (2*n+1)*Sum_{j=0..m-1} Fe(m,k)*(n*(n+1))^(m-1-j), m >= 2. Sums of even powers of the first n integers >0 as polynomials in u := n*(n+1) (falling powers of u). See bottom of p. 288 of the 1993 Knuth reference.
%D Ivo Schneider, Johannes Faulhaber 1580-1635, Birkhäuser Verlag, Basel, Boston, Berlin, 1993, ch. 7, pp. 131-159.
%H A. Dzhumadil'daev, D. Yeliussizov, <a href="http://cs.uwaterloo.ca/journals/JIS/VOL16/Yeliussizov/dzhuma6.html">Power sums of binomial coefficients</a>, Journal of Integer Sequences, 16 (2013), Article 13.1.4.
%H D. E. Knuth, <a href="http://dx.doi.org/10.1090/S0025-5718-1993-1197512-7">Johann Faulhaber and sums of powers</a>, Math. Comput. 203 (1993), 277-294.
%H W. Lang, <a href="/A093558/a093558.txt">First 10 rows and triangle with rational entries</a>.
%H D. Yeliussizov, <a href="https://web.archive.org/web/20160927104833/http://www.kazntu.kz/sites/default/files/20121221ND_Eleusizov.pdf">Permutation Statistics on Multisets</a>, Ph.D. Dissertation, Computer Science, Kazakh-British Technical University, 2012. [_N. J. A. Sloane_, Jan 03 2013]
%F a(n, m) = numerator(Fe(m, k), with Fe(m, k):=(m-k)*A(m, k)/(2*m*(2*m-1)) with Faulhaber numbers A(m, k):=A093556(m, k)/A093557(m, k) in Knuth's version. From the bottom of p. 288 of the 1993 Knuth reference.
%e [1]; [1,-1]; [1,-1,1]; [1,-1,1,-1]; ...
%e Numerators of [1/6]; [1/10,-1/30]; [1/14,-1/14,1/42]; [1/18,-1/9,1/10,-1/30]; ... (see W. Lang link)
%t a[m_, k_] := (-1)^(m-k)*Sum[Binomial[2*m, m-k-j]*Binomial[m-k+j, j]*((m-k-j)/(m-k+j))*BernoulliB[m+k+j], {j, 0, m-k}]; t[m_, k_] := (m-k)*a[m, k]/(2*m*(2*m-1)); Table[t[m, k] // Numerator, {m, 2, 12}, {k, 0, m-2}] // Flatten (* _Jean-François Alcover_, Mar 03 2014 *)
%K sign,frac,tabl,easy
%O 2,12
%A _Wolfdieter Lang_, Apr 02 2004
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