A093556 - OEIS

%I #29 Aug 28 2019 16:56:13

%S 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 -691,0,1,-8,112,-352,718,-280,140,0,1,-21,66,-293,4557,-3711,10851,

%U -10851,0,1,-40,217,-4516,2829,-26332,750167,-438670,219335,0,1,-33,506,-2585,7579,-198793,1540967,-627073

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

%C The companion triangle with the denominators is A093557.

%C 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 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 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 Ivo Schneider, Johannes Faulhaber 1580-1635, Birkhäuser Verlag, Basel, Boston, Berlin, 1993, ch. 7, pp. 131-159.

%H 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 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="/A093556/a093556.txt">First 10 rows and Faulhaber triangle with rational entries and examples</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(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 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 Triangle begins:

%e   [1];

%e   [1,0];

%e   [1,-1,0];

%e   [1,-4,2,0];

%e ...

%e Numerators of Knuth's Faulhaber triangle A(m,k):

%e   [1],

%e   [1, 0],

%e   [1, -1/2, 0],

%e   [1, -4/3, 2/3, 0],

%e   ...

%e A(m,m-1)=1 if m=1, else 0.

%e Edwards' Faulhaber triangle F^{-1}(m,l) = A(m,m-l)/m, for m>=2, l>=2:

%e   [1/2],

%e   [-1/6, 1/3],

%e   [1/6, -1/3, 1/4],

%e   [-3/10, 3/5, -1/2, 1/5],

%e   ...

%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}]; Flatten[ Table[ Numerator[a[m, k]], {m, 1, 11}, {k, 0, m-1}]] (* _Jean-François Alcover_, Oct 25 2011 *)

%Y 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 Cf. A103438.

%K sign,frac,tabl,easy

%O 1,8

%A _Wolfdieter Lang_, Apr 02 2004