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A093557 Triangle of denominators of coefficients of Faulhaber polynomials in Knuth's version.

%I A093557 #32 Aug 03 2025 16:26:26
%S A093557 1,1,1,1,2,1,1,3,3,1,1,2,1,2,1,1,1,2,1,1,1,1,6,15,3,15,30,1,1,1,3,3,3,
%T A093557 1,1,1,1,2,1,1,5,2,5,10,1,1,3,2,7,1,3,42,21,21,1,1,2,3,2,1,6,15,3,5,
%U A093557 10,1,1,1,5,3,10,5,15,5,5,1,1,1,1,6,3,2,3,3,7,1,1,14,21,42,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1
%N A093557 Triangle of denominators of coefficients of Faulhaber polynomials in Knuth's version.
%C A093557 The companion triangle with the numerators is A093556, where more information can be found.
%H A093557 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 A093557 Wolfdieter Lang, <a href="/A093557/a093557.txt">First 10 rows</a>.
%H A093557 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.
%F A093557 a(m, k) = denominator(A(m, k)) with recursion: A(m, 0)=1, A(m, k)=-(sum(binomial(m-j, 2*k+1-2*j)*A(m, j), j=0..k-1))/(m-k) if 0<= k <= m-1, else 0. From the 1993 Knuth reference, given in A093556, p. 288, eq.(*) with A^{(m)}_k = A(m, k).
%e A093557 Triangle begins:
%e A093557   [1];
%e A093557   [1,1];
%e A093557   [1,2,1];
%e A093557   [1,3,3,1];
%e A093557   ...
%e A093557 Denominators of [1]; [1,0]; [1,-1/2,0]; [1,-4/3,2/3,0]; ... (see W. Lang link in A093556.)
%t A093557 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}]; Flatten[ Table[ Denominator[a[m, k]], {m, 1, 14}, {k, 0, m-1}]] (* _Jean-François Alcover_, Oct 25 2011 *)
%o A093557 (PARI) T(n,k) = denominator((-1)^(n-k)*sum(j=0, n-k, binomial(2*n, n-k-j)*binomial(n-k+j,j)*(n-k-j)/(n-k+j) * bernfrac(n+k+j))); \\ _Michel Marcus_, Aug 03 2025
%Y A093557 Cf. A093556 (numerators).
%K A093557 nonn,frac,tabl,easy
%O A093557 1,5
%A A093557 _Wolfdieter Lang_, Apr 02 2004
%E A093557 More terms from _Michel Marcus_, Aug 03 2025