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A288245 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=7 data values.

%I A288245 #33 Jan 02 2025 22:19:26
%S A288245 1,7,-6,49,-63,15,343,-588,140,126,-20,2401,-5145,1225,2205,-175,-525,
%T A288245 15,16807,-43218,10290,27783,-1470,-8820,126,-2646,630,525,-6,117649,
%U A288245 -352947,84035,302526,-12005,-108045,1029,-64827,10290,8575,-49,15435,-441,-1225,1
%N A288245 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=7 data values.
%C A288245 Let SM_k = Sum( d_(t_1, t_2, t_3, ..., t_7)* eM_1^t_1 * eM_2^t_2 * ... * eM_7^t_7) summed over all length 7 integer partitions of k, i.e., 1*t_1 + 2*t_2 + 3*t_3 + ... + 7*t_7 = k, where SM_k are the averaged k-th power sum symmetric polynomials in 7 data (i.e., SM_k = S_k/7 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(7,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2, t_3, ..., t_7) form a triangle, with one row for each k value starting with k=1; the number of terms in successive rows is nondecreasing.
%C A288245 Row sums of positive entries give 1,7,64,609,5846,56161,... Row sums of negative entries are always 1 less than corresponding row sums of positive entries.
%H A288245 Gregory Gerard Wojnar, <a href="/A288245/a288245.java.txt">Java program</a>
%H A288245 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], 2107.
%e A288245 Triangular array begins...
%e A288245 1;
%e A288245 7,-6;
%e A288245 49,-63,15;
%e A288245 343,-588,140,126,-20;
%e A288245 2401,-5145,1225,2205,-175,-525,15;
%e A288245 16807,-43218,10290,27783,-1470,-8820,126,-2646,630,525,-6;
%e A288245 117649,-352947,84035,302526,-12005,-108045,1029,64827,10290,8575,-49,15435,-441,-1225,1;
%o A288245 (Java) // See Wojnar link.
%Y A288245 Cf. A028297 (m=2), A287768 (m=3), A288199 (m=4), A288211 (m=5), A288211 (m=6), A288188 (m=8). Also see Girard-Waring A210258.
%Y A288245 First entries of each row of triangle are powers of m=7, A000420.
%K A288245 sign,tabf
%O A288245 1,2
%A A288245 _Gregory Gerard Wojnar_, Jun 06 2017