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A093559 Triangle of denominators of coefficients of Faulhaber polynomials used for sums of even powers.

%I A093559 #22 Aug 09 2025 11:23:34
%S A093559 6,10,30,14,14,42,18,9,10,30,22,33,66,22,66,26,26,78,273,910,2730,30,
%T A093559 30,15,9,90,2,6,34,51,51,51,102,51,170,510,38,19,95,95,190,57,3990,
%U A093559 266,798,42,14,7,21,6,66,1386,693,110,330,46,138,46,23,230,690,345,23,230,46
%N A093559 Triangle of denominators of coefficients of Faulhaber polynomials used for sums of even powers.
%C A093559 The companion triangle with the numerators is A093558. See comment there.
%D A093559 Ivo Schneider, Johannes Faulhaber 1580-1635, Birkhäuser Verlag, Basel, Boston, Berlin, 1993, ch. 7, pp. 131-159.
%H A093559 A. Dzhumadil'daev and 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 A093559 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 A093559 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 A093559 a(n, m) = denominator(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 A093559 Triangle begins:
%e A093559   [6];
%e A093559   [10,30];
%e A093559   [14,14,42];
%e A093559   [18,9,10,30]; ...
%e A093559 Denominators of:
%e A093559   [1/6];
%e A093559   [1/10,-1/30];
%e A093559   [1/14,-1/14,1/42];
%e A093559   [1/18,-1/9,1/10,-1/30];
%e A093559   ... (see W. Lang link in A093558.)
%t A093559 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] // Denominator, {m, 2, 12}, {k, 0, m-2}] // Flatten (* _Jean-François Alcover_, Mar 03 2014 *)
%K A093559 nonn,frac,tabl,easy
%O A093559 2,1
%A A093559 _Wolfdieter Lang_, Apr 02 2004