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%I A288199 #32 Jul 04 2017 09:28:25
%S A288199 1,4,-3,16,-18,3,64,-96,16,18,-1,256,-480,80,180,-30,-5,1024,-2304,
%T A288199 384,1296,-288,-108,-24,9,12,4096,-10752,1792,8064,-2016,-1512,112,
%U A288199 252,-112,84,-7
%N A288199 Irregular triangle read by rows: mean version of Girard-Waring formula (cf. A210258), for m = 4 data values.
%C A288199 Let SM_k = Sum( d_(t_1, t_2, t_3, t_4)* eM_1^t_1 * eM_2^t_2 * eM_3^t_3*eM_4^t_4) summed over all length 4 integer partitions of k, i.e., 1*t_1 + 2*t_2 + 3*t_3 + 4*t_4 = k, where SM_k are the averaged k-th power sum symmetric polynomials in 4 data (i.e., SM_k = S_k/4 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(4,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2, t_3, t_4) 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 A288199 Row sums of positive entries give 1, 4, 19, 98, 516, 2725, 14400.
%C A288199 Row sums of negative entries are always 1 less than corresponding row sums of positive entries.
%H A288199 Gregory Gerard Wojnar, <a href="/A288199/a288199.java.txt">Java program</a>
%H A288199 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=4, p.23, arXiv:1706.08381 [math.GM], 2017.
%e A288199 Triangle begins:
%e A288199     1;
%e A288199     4,   -3;
%e A288199    16,  -18,   3;
%e A288199    64,  -96,  16,  18, -1;
%e A288199   256, -480,  80, 180, -5, -30;
%e A288199   ...
%e A288199 The first few rows describe:
%e A288199 Row 1: SM_1 = 1 eM_1;
%e A288199 Row 2: SM_2 = 4*(eM_1)^2 - 3*eM_2;
%e A288199 Row 3: SM_3 = 16*(eM_1)^3 - 18*eM_1*eM_2 + 3*eM_3;
%e A288199 Row 4: SM_4 = 64*(eM_1)^4 - 96*(eM_1)^2*eM_2 + 16*eM_1*eM_3 + 18*(eM_2)^2 - 1*eM_4;
%e A288199 Row 5: SM_5 = 256*(eM_1)^5 - 480*(eM_1)^3*eM_2 + 80*(eM_1)^2*eM_2 + 180*eM_1*(eM_2)^2 - 30*eM_2*eM_3 - 5*eM_1*eM_4.
%Y A288199 Cf. A210258, A028297 (m=2), A287768 (m=3), A288207 (m=5), A288211 (m=6), A288245 (m=7), A288188 (m=8).
%K A288199 sign,tabf,more
%O A288199 1,2
%A A288199 _Gregory Gerard Wojnar_, May 31 2017

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