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%I #18 Jun 24 2019 15:20:49
%S 1,1,-2,1,-3,3,1,-4,2,4,1,-5,5,5,-5,1,1,-6,9,6,-2,-12,3,1,-7,14,7,-7,
%T -21,7,7,1,-8,20,8,-16,-32,2,24,12,-8,1,-9,27,9,-30,-45,9,54,18,-9,
%U -27,3,1,-10,35,10,-50,-60,25,100,25,-2,-40,-60,15,10
%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 three indeterminates in terms of their elementary symmetric functions.
%C The length of row n is A001399(n), n >= 1.
%C The Girard-Waring formula for the power sum p(3,n) = x1^n + x2^2 + x3^n in terms of the elementary symmetric functions e_j(x1, x2, x3), for j=1, 2, 3, is given by Sum_{i=0..floor(n/3)} Sum_{j=0...floor((n-3*i)/2)} ((-1)^j)*n*(n - j - 2*i - 1)!/(i!*j!*(n - 2*j -3*i)!)*e_1^(n-3*i-2*j)*(e_2)^j*(e_3)^i, n >= 1 (the arguments of e_j have been omitted). See the W. Lang reference, Theorem 1, case N = 3, with r -> n.
%C This is an array using the partitions of n, in the reverse Abramowitz-Stegun order, with all partitions which have a part larger than 3 elininated. See row n of the array of Waring numbers A115131 read backwards, with these partitions omitted, and numerated with k from 1, 2, ..., A001399(n).
%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.
%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 >= 3.
%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 ...
%e -------------------------------------------------------------
%e 1: 1
%e 2: 1 -2
%e 3: 1 -3 3
%e 4: 1 -4 2 4
%e 5: 1 -5 5 5 -5
%e 6: 1 -6 9 6 -2 -12 3
%e 7: 1 -7 14 7 -7 -21 7 7
%e 8: 1 -8 20 8 -16 -32 2 24 12 -8
%e 9: 1 -9 27 9 -30 -45 9 54 18 -9 -27 3
%e 10: 1 -10 35 10 -50 -60 25 100 25 -2 -40 -60 15 10
%e ...
%e n = 4: x1^4 + x2^4 + x3^4 = (e_1)^4 - 4*(e_1)^2*e_2 + 2*(e_2)^2 + 4*e_1*e_3, with e_1 = x1 + x2 + x3, e_2 = x1*x2 + x1*x3 + x2*x^3 and e_3 = x1*x2*x3.
%Y Cf. A001399, A115131, A132460 (case N=2), A324602 (N=4).
%K sign,tabf,easy
%O 1,3
%A _Wolfdieter Lang_, May 03 2019
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