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A288211 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=6 data values.

%I A288211 #45 Jun 07 2024 11:15:15
%S A288211 1,6,-5,36,-45,10,216,-360,80,75,-10,1296,-2700,600,1125,-250,-75,5,
%T A288211 7776,-19440,4320,12150,-3600,-1125,-540,225,200,36,-1,46656,-136080,
%U A288211 30240,113400,-37800,-23625,2800,5250,-3780,3150,-350,252,-105,-7
%N A288211 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=6 data values.
%C A288211 Let SM_k = Sum( d_(t_1, t_2, ..., t_6)* eM_1^t_1 * eM_2^t_2 * ... * eM_6^t_6) summed over all length 6 integer partitions of k, i.e., 1*t_1 + 2*t_2 + 3*t_3 + ... + 6*t_6 = k, where SM_k are the averaged k-th power sum symmetric polynomials in 6 data (i.e., SM_k = S_k/6 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(6,k) with e_k being the k-th elementary symmetric polynomials.  The data d_(t_1, t_2, t_3, ..., t_6) form a triangle, with one row for each k value starting with k=1; the number of terms in successive rows is nondecreasing.
%C A288211 Row sums of positive entries give 1,6,46,371,3026,24707,201748. Row sums of negative entries are always 1 less than corresponding row sums of positive entries.
%H A288211 Gregory Gerard Wojnar, <a href="/A288211/a288211.java.txt">Java program</a>
%H A288211 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=6, p.24, arXiv:1706.08381 [math.GM], 2017.
%e A288211 Triangle begins:
%e A288211  1;
%e A288211  6,-5;
%e A288211  36,-45,10;
%e A288211  216,-360,80,75,-10;
%e A288211  1296,-2700,600,1125,-250,-75,5;
%e A288211  7776,-19440,4320,12150,-3600,-1125,-540,225,200,36,-1;
%e A288211  ...
%e A288211 Above represents:
%e A288211  SM_1 = eM_1;
%e A288211  SM_2 = 6*(eM_1)^2 - 5*eM_2;
%e A288211  SM_3 = 36*(eM_1)^3 - 45*eM_1*eM_2 + 10*eM_3;
%e A288211  SM_4 = 216*(eM_1)^4 - 360*(eM_1)^2*eM_2 + 80*eM_1*eM_3 + 75*(eM_2)^2 - 10*eM_4;
%e A288211  SM_5 = 1296*(eM_1)^5 - 2700*(eM_1)^3*eM_2 + 600*(eM_1)^2*eM_3 + 1125*eM_1*(eM_2)^2 - 250*eM_2*eM_3 - 75*eM_1*eM_4 + 5*eM_5;
%e A288211  ...
%o A288211 (Java) // See link.
%Y A288211 Cf. A028297 (m=2), A287768 (m=3), A288199 (m=4), A288207 (m=5), A288245 (m=7), A288188 (m=8). Also see A210258 Girard-Waring.
%Y A288211 First column of triangle is powers of m=6, A000400.
%K A288211 sign,tabf
%O A288211 1,2
%A A288211 _Gregory Gerard Wojnar_, Jun 06 2017