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A324602 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).

%I A324602 #16 Jun 24 2019 09:24:20
%S A324602 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 A324602 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 A324602 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 A324602 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 A324602 The length of row n is A001400(n), n >= 1.
%C A324602 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 A324602 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 A324602 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 A324602 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 A324602 The irregular triangle T(n, k) begins:
%e A324602 n\k 1   2  3  4   5   6  7  8   9  10  11  12  13   14   15  16  17 18 ...
%e A324602 -----------------------------------------------------------------------------
%e A324602 1:  1
%e A324602 2:  1  -2
%e A324602 3:  1  -3  3
%e A324602 4:  1  -4  2  4  -4
%e A324602 5:  1  -5  5  5  -5  -5
%e A324602 6:  1  -6  9  6  -2 -12 -6  3   6
%e A324602 7:  1  -7 14  7  -7 -21 -7  7   7  14  -7
%e A324602 8:  1  -8 20  8 -16 -32 -8  2  24  12  24  -8  -8  -16    4
%e A324602 9:  1  -9 27  9 -30 -45 -9  9  54  18  36  -9 -27  -27  -27   3  18  9
%e A324602 ...
%e A324602 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 A324602 ...
%e A324602 -----------------------------------------------------------------------------
%e A324602 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 A324602   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 A324602 Cf. A001400, A115131, A132460 (N=2), A325477 (N=3).
%K A324602 sign,tabf,easy
%O A324602 1,3
%A A324602 _Wolfdieter Lang_, May 03 2019