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A288207 Irregular triangle read by rows: mean version of Girard-Waring formula A210258, for m = 5 data values.

%I A288207 #48 Jun 11 2024 15:45:35
%S A288207 1,5,-4,25,-30,6,125,-200,40,40,-4,625,-1250,250,500,-25,-100,1,3125,
%T A288207 -7500,1500,4500,-150,-1200,6,-400,60,60,15625,-43750,8750,35000,-875,
%U A288207 -10500,35,-7000,700,700,1400,-14,-70
%N A288207 Irregular triangle read by rows: mean version of Girard-Waring formula A210258, for m = 5 data values.
%C A288207 Let SM_k = Sum( d_(t_1, t_2, ... , t_5)* eM_1^t_1 * eM_2^t_2 *...* eM_5^t_5) summed over all length 5 integer partitions of k, i.e., 1*t_1+2*t_2+...+5*t_5=k, where SM_k are the averaged k-th power sum symmetric polynomials in 5 data (i.e., SM_k = S_k/5 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(5,k) with e_k being the k-th elementary symmetric polynomials.  The data d_(t_1, t_2,... , t_5) form a triangle, with one row for each k value starting with k=1; the number of terms in successive rows is nondecreasing.
%H A288207 Gregory Gerard Wojnar, <a href="/A288207/a288207.java.txt">Java program</a>
%e A288207 Triangle begins:
%e A288207     1;
%e A288207     5,    -4;
%e A288207    25,   -30,   6;
%e A288207   125,  -200,  40,  40,   -4;
%e A288207   625, -1250, 250, 500, -100, -25, 1;
%e A288207   ...
%e A288207 Above represents:
%e A288207 SM_1 = 1*eM_1;
%e A288207 SM_2 = 5*(eM_1)^2 -4*eM_2;
%e A288207 SM_3 = 25*(eM_1)^3 - 30*eM_1*eM_2 + 6*eM_3;
%e A288207 SM_4 = 125*(eM_1)^4 - 200*(eM_1)^2*eM_2 + 40*eM_1*eM_3 + 40*(eM_2)^2 - 4*eM_4;
%e A288207 SM_5 = 625*(eM_1)^5 - 1250*(eM_1)^3*eM_2 + 250*(eM_1)^2*eM_3 + 500*eM_1*(eM_2)^2 - 100*eM_2*eM_3 - 25*eM_1*eM_4 + 1*eM_5;
%e A288207 ...
%o A288207 (Java) // See Java program link.
%Y A288207 Cf. A028297 (m=2), A287768 (m=3), A288199 (m=4), A288211 (m=6), A288245 (m=7), A288188 (m=8); A210258 Girard-Waring.
%Y A288207 First column of triangle are powers of m=5, A000351.
%K A288207 sign,tabf
%O A288207 1,2
%A A288207 _Gregory Gerard Wojnar_, Jun 06 2017