a(m,k) tabl head (triangle) for A093558: Numerators Faulhaber polynomials for even sums of powers m\k 0 1 2 3 4 5 6 7 8 9 2 1 0 0 0 0 0 0 0 0 0 3 1 -1 0 0 0 0 0 0 0 0 4 1 -1 1 0 0 0 0 0 0 0 5 1 -1 1 -1 0 0 0 0 0 0 6 1 -5 17 -5 5 0 0 0 0 0 7 1 -5 41 -236 691 -691 0 0 0 0 8 1 -7 14 -22 359 -7 7 0 0 0 9 1 -14 77 -293 1519 -1237 3617 -3617 0 0 10 1 -6 217 -1129 8487 -6583 750167 -43867 43867 0 11 1 -5 23 -470 689 -28399 1540967 -1254146 174611 -174611 etc. The companion triangle for the denominators is A093559: a(m,k) tabl head (triangle) for A093559 m\k 0 1 2 3 4 5 6 7 8 9 2 6 0 0 0 0 0 0 0 0 0 3 10 30 0 0 0 0 0 0 0 0 4 14 14 42 0 0 0 0 0 0 0 5 18 9 10 30 0 0 0 0 0 0 6 22 33 66 22 66 0 0 0 0 0 7 26 26 78 273 910 2730 0 0 0 0 8 30 30 15 9 90 2 6 0 0 0 9 34 51 51 51 102 51 170 510 0 0 10 38 19 95 95 190 57 3990 266 798 0 11 42 14 7 21 6 66 1386 693 110 330 etc. The triangle of rationals Fe(m,k):=A093558(m,k)/A093559(m,k) is m\k 0 1 2 3 4 5 6 7 8 9 2 2 0 0 0 0 0 0 0 0 0 3 3 -1 0 0 0 0 0 0 0 0 4 4 -4 4/3 0 0 0 0 0 0 0 5 5 -10 9 -3 0 0 0 0 0 0 6 6 -20 34 -30 10 0 0 0 0 0 7 7 -35 287/3 -472/3 691/5 -691/15 0 0 0 0 8 8 -56 224 -1760/3 2872/3 -840 280 0 0 0 9 9 -84 462 -1758 4557 -7422 32553/5 -10851/5 0 0 10 10 -120 868 -4516 16974 -131660/3 1500334/21 -438670/7 438670/21 0 11 11 -165 1518 -10340 53053 -198793 1540967/3 -2508292/3 3666831/5 -1222277/5 etc. The row polynomials with falling powers of u:=n*(n+1),starting with u^(m-1), multiplied with (2*n+1)/(2*m*(2*m-1)) give the sum of the 2*(m-1)-th power of the first n integers >0. For example: m=4: sum(j^6,j=1..n) = (4*(n*(n+1))^3 - 4*(n*(n+1))^2 +(4/3)*(n*(n+1))^1)*(2*n+1)/(2*4*7). Especially, for n=5: 1^6 + 2^6 + 3^6 + 4^6 + 5^6 = 20515 =(4*(5*6)^3 - 4*(5*6)^2 +(4/3)*(5*6)^1)*(2*5+1)/(2*4*7). #################################################################################################################### The actual triangle for the computation of the even powers of the first n numbers is (with 1/(2*m*(2*m-1)) factor) : m\k 0 1 2 3 4 5 6 7 8 9 2 1/6 0 0 0 0 0 0 0 0 0 3 1/10 -1/30 0 0 0 0 0 0 0 0 4 1/14 -1/14 1/42 0 0 0 0 0 0 0 5 1/18 -1/9 1/10 -1/30 0 0 0 0 0 0 6 1/22 -5/33 17/66 -5/22 5/66 0 0 0 0 0 7 1/26 -5/26 41/78 -236/273 691/910 -691/2730 0 0 0 0 8 1/30 -7/30 14/15 -22/9 359/90 -7/2 7/6 0 0 0 9 1/34 -14/51 77/51 -293/51 1519/102 -1237/51 3617/170 -3617/510 0 0 10 1/38 -6/19 217/95 -1129/95 8487/190 -6583/57 750167/3990 -43867/266 43867/798 0 11 1/42 -5/14 23/7 -470/21 689/6 -28399/66 1540967/1386 -1254146/693 174611/110 -174611/330 etc. With this triangle the row polynomials with falling powers of u:=n*(n+1), starting with u^(m-1), multiplied with (2*n+1) give the sum of the 2*(m-1)-th power of the first n integers >0. For example: m=4: sum(j^6,j=1..n) = ((1/14)*(n*(n+1))^3 - (1/14)*(n*(n+1))^2 + (1/42)*(n*(n+1))^1)*(2*n+1). Especially, for n=5: 1^6 + 2^6 + 3^6 + 4^6 + 5^6 = 20515 =((1/14)*(5*6)^3 - (1/14)*(5*6)^2 + (1/42)*(5*6))^1)*11. ############################################################################################################################### m\k 0 1 2 3 4 5 6 7 8 9 2 1/6 0 0 0 0 0 0 0 0 0 3 1/10 -1/30 0 0 0 0 0 0 0 0 4 1/14 -1/14 1/42 0 0 0 0 0 0 0 5 1/18 -1/9 1/10 -1/30 0 0 0 0 0 0 6 1/22 -5/33 17/66 -5/22 5/66 0 0 0 0 0 7 1/26 -5 41 -236 691 -691 0 0 0 0 8 1 -7 14 -22 359 -7 7 0 0 0 9 1 -14 77 -293 1519 -1237 3617 -3617 0 0 10 1 -6 217 -1129 8487 -6583 750167 -43867 43867 0 11 1 -5 23 -470 689 -28399 1540967 -1254146 174611 -174611 etc.