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A287768 Irregular triangle read by rows: mean version of Girard-Waring formula A210258, for m = 3 data values.

%I A287768 #52 Jan 02 2025 22:18:31
%S A287768 1,3,-2,9,-9,1,27,-36,4,6,81,-135,15,45,-5,243,-486,54,243,-36,-18,1,
%T A287768 729,-1701,189,1134,-189,-189,7,21,2187,-5832,648,4860,-864,-1296,36,
%U A287768 216,54,-8,6561,-19683,2187,19683,-3645,-7290,162,1458,729,-81,-81,1,19683,-65610,7290,76545,-14580,-36450,675,8100,6075,-540,-1080,10,-162,45
%N A287768 Irregular triangle read by rows: mean version of Girard-Waring formula A210258, for m = 3 data values.
%C A287768 Let SM_k = Sum( d_(t_1, t_2, t_3)* eM_1^t_1 * eM_2^t_2 * eM_3^t_3) summed over all length 3 integer partitions of k, i.e., 1*t_1+2*t_2+3*t_3=k, where SM_k are the averaged k-th power sum symmetric polynomials in 3 data (i.e., SM_k = S_k/3 where S_k are the k-th power sum symmetric polynomials, and where eM_k are the averaged k-th elementary symmetric polynomials, eM_k = e_k/binomial(3,k) with e_k being the k-th elementary symmetric polynomials.  The data d_(t_1, t_2, t_3) form an irregular triangle, with one row for each k value starting with k=1; "irregular" means that the number of terms in successive rows is nondecreasing.
%C A287768 The sum of the positive terms in successive rows appears to be A195350; row sums of negative terms is always 1 less than corresponding sum of positive terms.
%H A287768 Gregory Gerard Wojnar, <a href="/A287768/a287768.java.txt">Java program</a>
%H A287768 G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, <a href="http://arxiv.org/abs/1706.08381">Universal Peculiar Linear Mean Relationships in All Polynomials</a>, Table GW.n=3, p.22, arXiv:1706.08381 [math.GM], 2017.
%e A287768 Triangle begins:
%e A287768     1;
%e A287768     3,   -2;
%e A287768     9,   -9,  1;
%e A287768    27,  -36,  4,   6;
%e A287768    81, -135, 15,  45,  -5;
%e A287768   243, -486, 54, 243, -36, -18, 1;
%e A287768   ...
%e A287768 The first few rows describe:
%e A287768 Row 1: SM_1 = 1 eM_1;
%e A287768 Row 2: SM_2 = 3*(eM_1)^2 - 2*eM_2;
%e A287768 Row 3: SM_3 = 9*(eM_1)^3 - 9*eM_1*eM_2 + 1*eM_3;
%e A287768 Row 4: SM_4 = 27*(eM_1)^4 - 36*(eM_1)^2*eM_2 + 4*eM_1*eM_3 + 6*(eM_2)^2;
%e A287768 Row 5: SM_5 = 81*(eM_1)^5 - 135*(eM_1)^3*eM_2 + 15*(eM_1)^2*eM_3 + 45*eM_1*(eM_2)^2 - 5*eM_2*eM_3.
%o A287768 (Java) // See Wojnar link.
%Y A287768 Row sums of the positive terms appears to be A195350.
%Y A287768 First entries of row n is A000244(n).
%Y A287768 Second entries of row n, for n>1, is given by -n*3^(n-2).
%Y A287768 Third entries of row n, for n>2, is given by n*3^(n-4), A006234.
%Y A287768 Fourth entries of row n, for n>3, is given by n*(n-3)*3^(n-3)/2!.
%Y A287768 Fifth entries of row n, for n>4, is given by -n*(n-4)*3^(n-5)/1!.
%Y A287768 Corresponding sequences for different sized data multisets are:  A028297 (m=2), A288199 (m=4), A288207 (m=5), A288211 (m=6), A288245 (m=7), A288188 (m=8).
%Y A287768 Cf. A210258.
%K A287768 sign,tabf
%O A287768 1,2
%A A287768 _Gregory Gerard Wojnar_, May 31 2017