a(m,k) tabl head (triangle) for A092556, Faulhaber polynomials (numerator(A(m,k)) m\k 0 1 2 3 4 5 6 7 8 9 1 1 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 3 1 -1 0 0 0 0 0 0 0 0 4 1 -4 2 0 0 0 0 0 0 0 5 1 -5 3 -3 0 0 0 0 0 0 6 1 -4 17 -10 5 0 0 0 0 0 7 1 -35 287 -118 691 -691 0 0 0 0 8 1 -8 112 -352 718 -280 140 0 0 0 9 1 -21 66 -293 4557 -3711 10851 -10851 0 0 10 1 -40 217 -4516 2829 -26332 750167 -438670 219335 0 etc. The corresponding denominator triangle is A0925557 a(m,k) tabl head (triangle) for A093557, Faulhaber polynomials (denominator(A(m,k)) m\k 0 1 2 3 4 5 6 7 8 9 1 1 0 0 0 0 0 0 0 0 0 2 1 1 0 0 0 0 0 0 0 0 3 1 2 1 0 0 0 0 0 0 0 4 1 3 3 1 0 0 0 0 0 0 5 1 2 1 2 1 0 0 0 0 0 6 1 1 2 1 1 1 0 0 0 0 7 1 6 15 3 15 30 1 0 0 0 8 1 1 3 3 3 1 1 1 0 0 9 1 2 1 1 5 2 5 10 1 0 10 1 3 2 7 1 3 42 21 21 1 etc. and Knuth's Faulhaber triangle A^{m}_k = A(m,k):= A093556(m,k)/A093557(m,k) with rational entries is: m\k 0 1 2 3 4 5 6 7 8 9 1 1 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 3 1 -1/2 0 0 0 0 0 0 0 0 4 1 -4/3 2/3 0 0 0 0 0 0 0 5 1 -5/2 3 -3/2 0 0 0 0 0 0 6 1 -4 17/2 -10 5 0 0 0 0 0 7 1 -35/6 287/15 -118/3 691/15 -691/30 0 0 0 0 8 1 -8 112/3 -352/3 718/3 -280 140 0 0 0 9 1 -21/2 66 -293 4557/5 -3711/2 10851/5 -10851/10 0 0 10 1 -40/3 217/2 -4516/7 2829 -26332/3 750167/42 -438670/21 219335/21 0 etc. Faulhaber's polynomials (polynomials of row m in falling powers of u:=n*(n+1), starting with power m) divided by 2*m give the sum of the (2*m-1)-th power of the first n numbers. Example for sums of (2*m-1)-th power of the first n numbers for m=3: sum(j^5,j=1..n) = (1*(n*(n+1))^3 - (1/2)*(n*(n+1))^2)/(2*3) For example, with n=4: 1^5 + 2^5 + 3^5 + 4^5 = 1300 = ((4*5)^3 -(1/2)*(4*5)^2)/6 ######################################################################################################################### The triangle A(m,k)/(2*m) which is actually used for sums of odd powers of the first n numbers is: m\k 0 1 2 3 4 5 6 7 8 9 1 1/2 0 0 0 0 0 0 0 0 0 2 1/4 0 0 0 0 0 0 0 0 0 3 1/6 -1/12 0 0 0 0 0 0 0 0 4 1/8 -1/6 1/12 0 0 0 0 0 0 0 5 1/10 -1/4 3/10 -3/20 0 0 0 0 0 0 6 1/12 -1/3 17/24 -5/6 5/12 0 0 0 0 0 7 1/14 -5/12 41/30 -59/21 691/210 -691/420 0 0 0 0 8 1/16 -1/2 7/3 -22/3 359/24 -35/2 35/4 0 0 0 9 1/18 -7/12 11/3 -293/18 1519/30 -1237/12 3617/30 -3617/60 0 0 10 1/20 -2/3 217/40 -1129/35 2829/20 -6583/15 750167/840 -43867/42 43867/84 0 etc. Example for sums of (2*m-1)-th power of the first n numbers for m=4: sum(j^7,j=1..n) = (1/8)*(n*(n+1))^4 - (1/6)*(n*(n+1))^3 + (1/12)*(n*(n+1))^2 . E.g. for n=4: 1^7 + 2^7 + 3^7 +4^7 = 18700 = (1/8)*(4*5)^4 - (1/6)*(4*5)^3 + (1/12)*(4*5)^2 . #################################################################################################################### A. W. F. Edwards (see the 1986 ref. given in A093556 ) uses Faulhaber-polynomials with rising powers of u:=n*(n+1) starting with u^2, starts with m=2, and omits the zero entries in the main diagonal of the above triangles. Therefore his F^{-1} lower triangular matrix (his eq. 7) becomes F^{-1}(m,l) = A(m,m-l)/m (Edwards has a factor 1/2 in front of the F^{-1} matrix) in our notation. Therefore, his matrix F^{-1} is: m\l 2 3 4 5 6 7 8 9 10 2 1/2 0 0 0 0 0 0 0 0 3 -1/6 1/3 0 0 0 0 0 0 0 4 1/6 -1/3 1/4 0 0 0 0 0 0 5 -3/10 3/5 -1/2 1/5 0 0 0 0 0 6 5/6 -5/3 17/12 -2/3 1/6 0 0 0 0 7 -691/210 691/105 -118/21 41/15 -5/6 1/7 0 0 0 8 35/2 -35 359/12 -44/3 14/3 -1 1/8 0 0 9 -3617/30 3617/15 -1237/6 1519/15 -293/9 22/3 -7/6 1/9 0 10 43867/42 -43867/21 750167/420 -13166/15 2829/10 -2258/35 217/20 -4/3 1/10 etc. This reappears as subtriangle in: Ira M. Gessel and X. G. Viennot, Determinants, Paths, and Plane Partitions, preprint, 1989. Their f(n,k)= A065551(n,k)/A065553(n,k), and Edwards' F^{-1}(m,l) = f(m-1,l-1) = A(m,m-l)/m, or f(n,k) = A(n+1,n-k)/(n+1). Example for sums of (2*m-1)-th power of the first n numbers for m=4 (remember the extra 1/2 factor): sum(j^7,j=1..n) = (1/2)* ((1/6)*(n*(n+1))^2 - (1/3)*(n*(n+1))^3 + (1/4)*(n*(n+1))^4) . E.g. for n=4: 1^7 + 2^7 + 3^7 +4^7 = 18700 = (1/2)*((1/6)*(4*5)^2 - (1/3)*(4*5)^3 + (1/4)*(4*5)^4) . ##########################################################################################################