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A308684 Partition array T(n, k) for the coefficients of the n-th power sums of the second elementary symmetric function in terms of the elementary symmetric functions.

%I A308684 #8 Aug 20 2021 22:50:47
%S A308684 1,2,-2,1,3,-3,-3,3,3,-3,1,4,-4,-4,-4,6,4,8,-8,-4,4,-4,4,2,-4,1,5,-5,
%T A308684 -5,-5,-5,10,5,10,10,-15,5,-15,5,5,-5,-15,10,5,10,-5,-5,-5,5,5,5,-5,5,
%U A308684 5,-5,1
%N A308684 Partition array T(n, k) for the coefficients of the n-th power sums of the second elementary symmetric function in terms of the elementary symmetric functions.
%C A308684 The length of row n is A209816(n) (number of partitions of 2*n with at most n parts).
%C A308684 This is a generalization of the Girard-Waring array A115131.
%C A308684 In A324254 the general definition psigma(n, r) has been given for the r-th power sum of the n-th elementary symmetric function. There it is given in terms of the ordinary power sums {ps(j*r)}_{j=1..n}. Here psigma(2, n) = (1/2)*(-ps(2*n) + (ps(n))^2) is considered (see row n = 2 in A324254), and it is written in terms of elementary symmetric functions e_k(x1, x2, ...x_N), using the Girard-Waring formula for power sums ps. The number N >= 1 of indeterminates is suppressed in the notations.
%H A308684 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP?Res=150&amp;Page=831&amp;Submit=Go">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%F A308684 psigma(2, n) = Sum_{k=1.. A209816(n)} T(n, k)*Product_{j=1..2*n} (e_j)^a(2*n,k,j), for n >= 1, if the k-th partition of 2*n (in Abramowitz-Stegun order) is Product_{j=1..2*n} j^a(2*n,k,j). Here the elemntary symmetric functions are e_j = e_j^{(N)} for N indeterminates x_1, ..., x_N, for any N >= 1.
%e A308684 The irregular triangle (partition array) T(n, k)  begins:
%e A308684 n\k 1  2  3  4  5   6  7  8   9  10  11  12 13 14 15 ...
%e A308684 -------------------------------------------------------------------------------------------
%e A308684 1:  1
%e A308684 2:  2 -2  1
%e A308684 3:  3 -3 -3  3  3  -3  1
%e A308684 4:  4 -4 -4 -4  6   4  8 -8  -4   4  -4   4  2 -4  1
%e A308684 ...
%e A308684 n = 5: [[5], [-5, -5, -5, -5, 10], [5, 10, 10, -15, 5, -15, 5, 5], [-5, -15, 10, 5, 10, -5, -5, -5, 5], [5, 5, -5, 5, 5, -5, 1]];
%e A308684 n = 6: [[6],[-6, -6, -6, -6, -6, 15], [6, 12, 12, 12, -24, 6, 12. -24, 6, -12, 12, 2], [-6, -18, -18, 18, 9, -18, 36, 0, 0, -18, 6, -18, 9, 0, 3], [6, 24, -12, -12, -18, 0, 0, 18, 12, 0, -12, -6, 6], [-6, 6, 6, 3,-6,-12, -2, 6, 9, -6, 1]];
%e A308684 n = 7: [[7], [-7, -7, -7, -7, -7, -7, 21] , [7, 14, 14, 14, 14, -35, 7, 14, 14, -35, 7, 7, -35, 14, 7, 7], [-7, -21, -21, -21, 28, 14, -21,-42, 56, 7, 28, 7, -21, -21, -7, 28, -21, 14, -21, 7, -7, -7, 14],  [7, 28, 28, -21, -21, 42, -63, -14, -7, -7, 35, 14, -21, 35, -14, 35, -14, -21, 7, -21, 14, -7, 7], [-7, -35, 14, 14, 7, 28, 7, 7, -21, -21,-21,-14, -7, 35, 7, 14, 7, -21, -7, 7], [7, -7, -7, -7, 7, 14, 7, 7, -7, -21, -7, 7, 14, -7, 1]]:
%e A308684 Brackets combine terms belonging to the same number of parts.
%e A308684 ...
%e A308684 n = 3: psigma(2, 3) := Sum_{1<= i1 < i2 <= N} (x_{i1}*x_{i2})^3 = (1/2)*(-ps(2*3) + (ps(3))^2) = 3*e_6 - 3*e_1*e_5 - 3*e_2*e_4 + 3*(e_3)^2 + 3*(e_1)^2*e_4 - 3*e_1*e_2*e_3 + (e_2)^3. This becomes an identity if the e_j are written in terms of the indeterminates x_1, ..., x_N, for any N >= 1.
%Y A308684 Cf. A115121, A324254 (psigma(2, n) in terms of power sums).
%K A308684 sign,tabf
%O A308684 1,2
%A A308684 _Wolfdieter Lang_, Jul 08 2019