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A288188 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=8 data values.

%I A288188 #34 Jan 02 2025 22:18:58
%S A288188 1,8,-7,64,-84,21,512,-896,224,196,-35,4096,-8960,2240,3920,-350,-980,
%T A288188 35,32768,-86016,21504,56448,-3360,-18816,336,-5488,1470,1176,-21,
%U A288188 262144,-802816,200704,702464,-31360,-263424,3136,-153664,27440,21952,-196,38416,-1372,-3430,7
%N A288188 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=8 data values.
%C A288188 Let SM_k = Sum( d_(t_1, t_2, t_3, ..., t_8)* eM_1^t_1 * eM_2^t_2 * ...*eM_8^t_8) summed over all length 8 integer partitions of k, i.e., 1*t_1+2*t_2+3*t_3+...+8*t_8=k, where SM_k are the averaged k-th power sum symmetric polynomials in 8 data (i.e., SM_k = S_k/8 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(8,k) with e_k being the k-th elementary symmetric polynomials.  The data d_(t_1, t_2, t_3, ..., t_8) form a triangle, with one row for each k value starting with k=1; the number of terms in successive rows is nondecreasing.
%C A288188 Row sums of positive entries give: 1,8,85,932,10291,114878,... Row sums of negative entries are always 1 less than corresponding row sums of positive entries.
%H A288188 Gregory Gerard Wojnar, <a href="/A288188/a288188.java.txt">Java program</a>
%H A288188 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>, pp. 22-24, arXiv:1706.08381 [math.GM], 2017.
%e A288188 Triangle begins
%e A288188      1;
%e A288188      8,    -7;
%e A288188     64,   -84,   21;
%e A288188    512,  -896,  224,  196,  -35;
%e A288188   4096, -8960, 2240, 3920, -350, -980, 35;
%e A288188   ...
%o A288188 (Java) // See Wojnar link.
%Y A288188 Cf. A028297 (m=2), A287768 (m=3), A288199 (m=4), A288207 (m=5), A288211 (m=6), A288245 (m=7). See Girard-Waring A210258. T(n,1)=8^(n-1)=A001018(n).
%K A288188 sign,tabf
%O A288188 1,2
%A A288188 _Gregory Gerard Wojnar_, Jun 16 2017