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A093556 Triangle of numerators of coefficients of Faulhaber polynomials in Knuth's version.

%I A093556 #42 Aug 03 2025 16:26:21
%S A093556 1,1,0,1,-1,0,1,-4,2,0,1,-5,3,-3,0,1,-4,17,-10,5,0,1,-35,287,-118,691,
%T A093556 -691,0,1,-8,112,-352,718,-280,140,0,1,-21,66,-293,4557,-3711,10851,
%U A093556 -10851,0,1,-40,217,-4516,2829,-26332,750167,-438670,219335,0,1,-33,506,-2585,7579,-198793,1540967,-627073,1222277,-1222277,0
%N A093556 Triangle of numerators of coefficients of Faulhaber polynomials in Knuth's version.
%C A093556 The companion triangle with the denominators is A093557.
%C A093556 In the 1986 Edwards reference, eq. 7, p. 453, the lower triangular matrix F^{-1} is obtained from F^{-1}(m,l) = A(m,m-l)/m with m >= 2, l >= 2. See the W. Lang link for this triangle.
%C A093556 Sum_{j=1..n} j^(2*m-1) = Sum_{k=0..m-1} A(m,k)*u^(m-k)/(2*m), with u:=n*(n+1), A(m,k):= A093556(m,k)/ A093557(m,k) and m=1,2,... (Faulhaber's m-th row polynomial in falling powers of u:=n*(n+1), divided by 2*m, gives the sum of the (2*m-1)-th power of the first n integers > 0. See the W. Lang link for the Faulhaber triangle.)
%C A093556 Sum_{j=1..n} j^(2*(m-1)) = (2*n+1)*Sum_{j=0..m-1} (m-j)*A(m,j)*(n*(n+1))^(m-1-j)/(2*m*(2*m-1)), with u:=n*(n+1) and m >= 2. Sum of the even powers of the first n integers > 0. From the bottom of p. 288 of the 1993 Knuth reference with A^{(m)}_k = A(m,k). See also A093558 with A093559.
%D A093556 Ivo Schneider, Johannes Faulhaber 1580-1635, Birkhäuser Verlag, Basel, Boston, Berlin, 1993, ch. 7, pp. 131-159.
%H A093556 A. W. F. Edwards, <a href="http://www.jstor.org/stable/2323466">A quick route to sums of powers</a>, Amer. Math. Monthly 93 (1986) 451-455.
%H A093556 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 A093556 Wolfdieter Lang, <a href="/A093556/a093556.txt">First 10 rows and Faulhaber triangle with rational entries and examples</a>.
%H A093556 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 A093556 a(m, k) = numerator(A(m, k)) with recursion: A(m, 0)=1, A(m, k) = -(Sum_{j=0..k-1} binomial(m-j, 2*k+1-2*j)*A(m, j))/(m-k) if 0 <= k <= m-1, otherwise 0. From the Knuth 1993 reference, p. 288, eq.(*) with A^{(m)}_k = A(m, k).
%F A093556 A(m, k) = ((-1)^(m-k))*Sum_{j=0..m-k} binomial(2*m, m-k-j)*binomial(m-k+j, j)*((m-k-j)/(m-k+j))*Bernoulli(m+k+j). From the Knuth 1993 reference, p. 289, last eq. with A^{(m)}_k = A(m, k). Attributed to I. M. Gessel and X. G. Viennot (see A065551 for the 1989 reference). For Bernoulli numbers see A027641 with A027642.
%e A093556 Triangle begins:
%e A093556   [1];
%e A093556   [1,0];
%e A093556   [1,-1,0];
%e A093556   [1,-4,2,0];
%e A093556 ...
%e A093556 Numerators of Knuth's Faulhaber triangle A(m,k):
%e A093556   [1],
%e A093556   [1, 0],
%e A093556   [1, -1/2, 0],
%e A093556   [1, -4/3, 2/3, 0],
%e A093556   ...
%e A093556 A(m,m-1)=1 if m=1, else 0.
%e A093556 Edwards' Faulhaber triangle F^{-1}(m,l) = A(m,m-l)/m, for m>=2, l>=2:
%e A093556   [1/2],
%e A093556   [-1/6, 1/3],
%e A093556   [1/6, -1/3, 1/4],
%e A093556   [-3/10, 3/5, -1/2, 1/5],
%e A093556   ...
%t A093556 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[ Numerator[a[m, k]], {m, 1, 11}, {k, 0, m-1}]] (* _Jean-François Alcover_, Oct 25 2011 *)
%o A093556 (PARI) T(n,k) = numerator((-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 A093556 Cf. A093557 (denominators).
%Y A093556 Cf. A065551 and A065553 for Ira M. Gessel's and X. G. Viennot's version of Faulhaber triangle which is Edwards' Faulhaber triangle augmented with a first row and first column.
%Y A093556 Cf. A027641, A027642, A093558, A093559, A103438.
%K A093556 sign,frac,tabl,easy
%O A093556 1,8
%A A093556 _Wolfdieter Lang_, Apr 02 2004
%E A093556 More terms from _Michel Marcus_, Aug 03 2025