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%I #16 Jun 24 2019 09:24:20
%S 1,1,-2,1,-3,3,1,-4,2,4,-4,1,-5,5,5,-5,-5,1,-6,9,-2,6,-12,3,-6,6,1,-7,
%T 14,-7,7,-21,7,7,-7,14,-7,1,-8,20,-16,2,8,-32,24,12,-8,-8,24,-8,-16,4,
%U 1,-9,27,-30,9,9,-45,54,-9,18,-27,3,-9,36,-27,-27,18,9,1,-10,35,-50,25,-2,10,-60,100,-40,25,-60,15,10,-10,50,-60,10,-40,60,-10,15,-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 four indeterminates in terms of their elementary symmetric functions (reverse Abramowitz-Stegun order of partitions).
%C The length of row n is A001400(n), n >= 1.
%C The Girard-Waring formula for the power sum p(4,n) := Sum_{j=1..4} (x_j)^n in terms of the elementary symmetric functions e_j(x_1, x_2, x_3, x_4), for j = 1, 2, 3, 4, is given by Sum_{i1=0..floor(n/4)} Sum_{i2=0...floor((n-4*i1)/3)} Sum_{i3=0...floor((n-4*i1-3*i2)/2)} ((-1)^(i1 + i3))*n*(n-1-i3-2*i2-3*i1)!/(i1!*i2!*i3!*(n-2*i3-3*i2-4*i1)!)*e_1^(n-2*i3-3*i2-4*i1)*(e_2)^i3*(e_3)^i2*(e_4)^i1, n >= 1 (the arguments of e_j have been omitted). See the W. Lang reference, Theorem 1, case N = 4, with r -> n.
%C This is an array using the partitions of n, in the reverse Abramowitz-Stegun order, with all partitions with a part >= 5 eliminated. See row n of the array of Waring numbers A115131, read backwards, with these partitions omitted.
%H Wolfdieter Lang, <a href="https://doi.org/10.1016/S0377-0427(97)00240-9">On sums of powers of zeros of polynomials</a>, J. Comp. Appl. Math. 89 (1998) 237-356.
%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 >= 5.
%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 ...
%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
%e 6: 1 -6 9 6 -2 -12 -6 3 6
%e 7: 1 -7 14 7 -7 -21 -7 7 7 14 -7
%e 8: 1 -8 20 8 -16 -32 -8 2 24 12 24 -8 -8 -16 4
%e 9: 1 -9 27 9 -30 -45 -9 9 54 18 36 -9 -27 -27 -27 3 18 9
%e ...
%e n = 10: 1 -10 35 10 -50 -60 -10 25 100 25 50 -2 -40 -60 -60 -40 15 10 10 60 15 -10 -10.
%e ...
%e -----------------------------------------------------------------------------
%e Row n = 5: p(4,5) = x_1^5 + x_2^5 + x_3^5 + x_4^5 = 1*e_1^5 - 5* e_1^3*e_2 + 5*e_1*e_2^2 + 5*e_1^2*e_3 - 5*e_2*e_3 - 5*e_1*e_4,
%e with e_1 = Sum_{j=1..4} x_j, e_2 = x1*(x_2 + x_3 + x_4) + x_2*(x_3 + x_4) + x_3*x_4, e_3 = x_1*x_2*x_3 + x_1*x_2*x_4 + x_2*x_3*x_4, e_4 = Product_{i=1..4} x_j.
%Y Cf. A001400, A115131, A132460 (N=2), A325477 (N=3).
%K sign,tabf,easy
%O 1,3
%A _Wolfdieter Lang_, May 03 2019
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