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A210258 The coefficients of the Girard-Waring formula; irregular array T(n,k), read by rows, for n >= 1 and 1 <= k <= A000041(n).

%I A210258 #71 Jul 18 2025 09:33:23
%S A210258 1,1,-2,1,-3,3,1,-4,4,2,-4,1,-5,5,5,-5,-5,5,1,-6,6,9,-6,-12,6,-2,6,3,
%T A210258 -6,1,-7,7,14,-7,-21,7,-7,14,7,-7,7,-7,-7,7,1,-8,8,20,-8,-32,8,-16,24,
%U A210258 12,-8,24,-16,-16,8,2,-8,-8,8,8,8,-8,1,-9,9,27,-9,-45
%N A210258 The coefficients of the Girard-Waring formula; irregular array T(n,k), read by rows, for n >= 1 and 1 <= k <= A000041(n).
%C A210258 Assume we have N <= n variables x_1, x_2, ..., x_N, and let S_n^{(N)} = x_1^n + x_2^n + ... + x_N^n be the n-th power sum of these variables. Following Gould (1999, p. 135), define the elementary symmetric polynomials e_1^{(N)}, e^2^{(N)}, ..., e_N^{(N)}, where e_j^{(N)} is the sum of all products of x_1, x_2, ..., x_N taken j at a time.
%C A210258 Since N <= n, without loss of generality, we may assume we have the extra variables x_{N+1}, ..., x_n, define e_1^{(n)}, e_2^{(n)}, ..., e_N^{(n)}, e_{N+1}^{(n)}, ..., e_n^{(n)}, and then set x_{N+1} = x_{N+2} = ... = x_n = 0. In such a case, we get e_j^{(N)} = e_j^{(n)} for j = 1..N, and e_j^{(n)} = 0 for j = (N+1)..n. Thus, we may drop the superscripts N and n on power sum and on the elementary symmetric polynomials and write S_n, and e_1, e_2, ..., e_n with the understanding that e_{N+1} = ... = e_n = 0. (See also the comments for A115131.)
%C A210258 The numbers T(n,k) are the coefficients of the power sum expansion in terms of the elementary symmetric polynomials, which is the Girard-Waring formula. That is, S_n = Sum(c(t_1,...,t_n) * e_1^t_1 * e_2^t_2 *...* e_n^t_n), summed over integer partitions of n, t_1 + 2*t_2 + ... + n*t_n = n with t_i >= 0 for i = 1..n. Here c(t_1,t_2,...,t_n) = n * (-1)^(t_2 + t_4 + ... + t_{2*floor(n/2)}) * (t_1 + ... + t_n - 1)!/(t_1!*...*t_n!).
%C A210258 Given a partition (t_1,...,t_n) of n (as defined above), we may use only those j's in {1,...,n} for which t_j > 0 and write the partition in the usual notation by repeating each such j t_j times (and place these j's in a non-descending order). E.g., the partition 1*3 + 2*1 + 3*0 + 4*0 + 5*0 of 5 can be written as [1,1,1,2]. Similarly, the partition 1*5 + 2*0 + 3*0 + 4*0 + 5*0 of 5 can be written as [1,1,1,1,1].
%C A210258 Instead of using the Abramowitz-Stegun order of partitions (as it is done in A115131), we use the usual notation for partitions (in terms of the positive integers that add up to n) and order them in lexicographic order. Unfortunately, this order of partitions does not correspond to any of the orders in the web link below.
%C A210258 If we let a(m) be the m-th term of the array, read as sequence, then
%C A210258 a(1) = T(1,1) = c([1]) = c(1),
%C A210258 a(2) = T(2,1) = c([1,1]) = c(2,0) with sign(T(2,1)) = (-1)^0 = 1,
%C A210258 a(3) = T(2,2) = c([2]) = c(0,1) with sign(T(2,2)) = (-1)^1 = -1,
%C A210258 a(4) = T(3,1) = c([1,1,1]) = c(3,0,0) with sign(T(3,1)) = (-1)^0 = 1,
%C A210258 a(5) = T(3,2) = c([1,2]) = c(1,1,0) with sign(T(4,1)) = (-1)^1 = -1,
%C A210258 a(6) = T(3,3) = c([3]) = c(0,0,1) with sign(T(3,3)) = (-1)^0 = 1,
%C A210258 a(7) = T(4,1) = c([1,1,1,1]) = c(4,0,0,0) with sign(T(4,1)) = (-1)^(0+0) = 1,
%C A210258 a(8) = T(4,2) = c([1,1,2]) = c(2,1,0,0) with sign(T(4,2)) = (-1)^(1+0) = -1,
%C A210258 a(9) = T(4,3) = c([1,3]) = c(1,0,1,0) with sign(T(4,3)) = (-1)^(0+0) = 1,
%C A210258 a(10) = T(4,4) = c([2,2]) = c(0,2,0,0) with sign(T(4,4)) = (-1)^(2+0) = 1,
%C A210258 a(11) = T(4,5) = c([4]) = c(0,0,0,1) with sign(T(4,5)) = (-1)^(0+1) = -1,
%C A210258 a(12) = T(5,1) = c([1,1,1,1,1]) = c(5,0,0,0,0),
%C A210258 a(13) = T(5,2) = c([1,1,1,2]) = c(3,1,0,0,0),
%C A210258 a(14) = T(5,3) = c([1,1,3]) = c(2,0,1,0,0),
%C A210258 a(15) = T(5,4) = c([1,2,2]) = c(1,2,0,0,0),
%C A210258 a(16) = T(5,5) = c([1,4]) = c(1,0,0,1,0),
%C A210258 a(17) = T(5,6) = c([2,3]) = c(0,1,1,0,0),
%C A210258 a(18) = T(5,7) = c([5]) = c(0,0,0,0,1), ...
%C A210258 (ascending ordered compositions in lexicographic order).
%H A210258 Kahtan H. Alzubaidy, <a href="https://www.maplesoft.com/applications/view.aspx?SID=153837">Symmetric polynomials by Maple</a>, 2015 (requires Maple 13).
%H A210258 Henri W. Gould, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/37-2/gould.pdf">The Girard-Waring power sum formulas for symmetric functions and Fibonacci sequences</a>, Fibonacci Quart. 37(2) (1999), 135-140. [He uses a signed version of the elementary symmetric polynomials: a_j = (-1)^j*e_j. This is why here we have (-1)^(t_2 + t_4 + ... + t_{2*floor(n/2)}) in the formula for c(t_1,...,t_n) rather than (-1)^(t_1 + ... + t_n).]
%H A210258 OEIS, <a href="https://oeis.org/wiki/Orderings_of_partitions">Orderings of partitions</a>.
%H A210258 Wikipedia, <a href="https://en.wikipedia.org/wiki/Newton%27s_identities">Newton's identities</a>.
%H A210258 Gregory Gerard Wojnar, Daniel Sz. Wojnar, and Leon Q. Brin, <a href="https://arxiv.org/abs/1706.08381">Universal Peculiar Linear Mean Relationships in All Polynomials</a>, arXiv:1706.08381 [math.GM], 2017. See p. 4.
%e A210258 Array T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
%e A210258   S_1: 1;
%e A210258   S_2: 1, -2;
%e A210258   S_3: 1, -3, 3;
%e A210258   S_4: 1, -4, 4,  2, -4;
%e A210258   S_5: 1, -5, 5,  5, -5,  -5, 5;
%e A210258   S_6: 1, -6, 6,  9, -6, -12, 6, -2,  6, 3, -6;
%e A210258   S_7: 1, -7, 7, 14, -7, -21, 7, -7, 14, 7, -7, 7, -7, -7, 7;
%e A210258   ...
%e A210258 With N = n = 6, we have S_6 = 1*(e_1)^6 - 6*(e_1)^4*(e_2) + 6*(e_1)^3*(e_3) + 9*(e_1)^2*(e_2)^2 - 6*(e_1)^2*(e_4) - 12*(e_1)*(e_2)*(e_3) + 6*(e_1)*(e_5) - 2*(e_2)^3 + 6*(e_2)*(e_4) + 3*(e_3)^2 - 6*(e_6) = Sum_{i = 1..6} x_i^6.
%e A210258 If N = 4 < n = 6, we set e_5 = e_6 = 0 in the above expression, and we get that S_6 = 1*(e_1)^6 - 6*(e_1)^4*(e_2) + 6*(e_1)^3*(e_3) + 9*(e_1)^2*(e_2)^2 - 6*(e_1)^2*(e_4) - 12*(e_1)*(e_2)*(e_3) - 2*(e_2)^3 + 6*(e_2)*(e_4) + 3*(e_3)^2 = Sum_{i = 1..4} x_i^6.
%Y A210258 Cf. A115131 (Abramowitz-Stegun order of partitions).
%K A210258 sign,tabf
%O A210258 1,3
%A A210258 _Mircea Merca_, Mar 19 2012
%E A210258 Various sections edited by _Petros Hadjicostas_, Dec 14 2019