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%I #13 Jun 24 2019 17:48:47
%S 1,1,-2,1,-3,3,1,-4,2,4,-4,1,-5,5,5,-5,-5,5,1,-6,9,6,-2,-12,-6,3,6,6,
%T 1,-7,14,7,-7,-21,-7,7,7,14,7,-7,-7,1,-8,20,8,-16,-32,-8,2,24,12,24,8,
%U -8,-8,-16,-16,4,8,1,-9,27,9,-30,-45,-9,9,54,18,36,9,-9,-27,-27,-27,-27,3,18,9,9,18,-9,1,-10,35,10,-50,-60,-10,25,100,25,50,10,-2,-40,-60,-60,-40,-40,15,10,10,60,30,15,30,-10,-10,-20,-20,5
%N Irregular triangle read by rows: T(n, k) gives the coefficients of the Girard-Waring formula for the sum of n-th power of five indeterminates in terms of their elementary symmetric functions (reverse Abramowitz-Stegun order of partitions).
%C The length of row n is A001401(n), n >= 1.
%C The Girard-Waring formula for the power sum p(5,n) = Sum_{j=1..5} (x_j)^n in terms of the elementary symmetric functions e_j(x_1, x_2, x_3, x_4), for j = 1, 2 ,..., 5 is given in the W. Lang reference, Theorem 1, in an explicitly nested four sums version. See also the summary link, for N = 5 (there sigma_j^{(N)} -> e_j here).
%C In this array the partitions of n, with all partitions with a part >= 6 omitted, are used. Here the partitions appear in the reverse Abramowitz-Stegun order. See row n of the array of Waring numbers A115131, read backwards, with the entries corresponding to these omitted partitions.
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP?Res=150&Page=821">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]
%H Wolfdieter Lang, <a href="http://dx.doi.org/10.1016/S0377-0427(97)00240-9">On sums of powers of zeros of polynomials</a>, J. Comp. Appl. Math. 89 (1998) 237-256; Theorem 1.
%H Wolfdieter Lang, <a href="/A307449/a307449.pdf">Nested sum version of the Girard-Waring formula (a summary)</a>
%F T(n, k) is the k-th coefficient of the Waring number partition array A115131(n, m) (k there is replaced here by m), read backwards, omitting all partitions which have a part >= 6.
%e The irregular triangle T(n, k) begins:
%e n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
%e -----------------------------------------------------------------------------
%e 1: 1
%e 2: 1 -2
%e 3: 1 -3 3
%e 4: 1 -4 2 4 -4
%e 5: 1 -5 5 5 -5 -5 5
%e 6: 1 -6 9 6 -2 -12 -6 3 6 6
%e 7: 1 -7 14 7 -7 -21 -7 7 7 14 7 -7 -7
%e 8: 1 -8 20 8 -16 -32 -8 2 24 12 24 8 -8 -8 -16 -16 4 8
%e 9: 1 -9 27 9 -30 -45 -9 9 54 18 36 9 -9 -27 -27 -27 -27 3 18 9 9 18 -9
%e .
%e .
%e .
%e n = 10: 1 -10 35 10 -50 -60 -10 25 100 25 50 10 -2 -40 -60 -60 -40 -40 15 10 10 60 30 15 30 -10 -10 -20 -20 5.
%e ...
%e ------------------------------------------------------------------------------
%e Row n = 6: x_1^6 + x_2^6 + x_3^6 + x_4^6 + x_5^6 = 1*e_1^6 - 6*e_1^4*e_2 + 9*e_1^2*e_2^2 + 6*e_1^3*e_3 - 2*e_2^3 - 12*e_1*e_2*e_3 - 6*e_1^2*e_4 + 3*e_3^2 + 6*e_2*e_4 + 6*e_1*e_5, with e_1 = Sum_{j=1..5} x_j, e_2 = x1*(x_2 + x_3 + x_4 + x_5) + x_2*(x_3 + x_4 + x_5) + x_3*(x_4 + x_5) + x_4*x_5, e_3 = x_1*x_2*x_3 + x_1*x_2*x_4 + x_1*x_2*x_5 + x_2*x_3*x_4 + x_2*x_3*x_5 + x_2*x_4*x_5 + x_3*x_4*x_5, e_4 = x_1*x_2*x_3*x_4 + x_1*x_2*x_3*x_5 + x_1*x_2*x_4*x_5 + x_1*x_3*x_4*x_5 + x_2*x_3*x_4*x_5, e_5 = Product_{i=1..5} x_j.
%Y Cf. A001401, A115131, A132460 (N=2), A325477 (N=3), A324602 (N=4).
%K sign,tabf
%O 1,3
%A _Wolfdieter Lang_, May 14 2019
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