A287768 - OEIS

%I #52 Jan 02 2025 22:18:31

%S 1,3,-2,9,-9,1,27,-36,4,6,81,-135,15,45,-5,243,-486,54,243,-36,-18,1,

%T 729,-1701,189,1134,-189,-189,7,21,2187,-5832,648,4860,-864,-1296,36,

%U 216,54,-8,6561,-19683,2187,19683,-3645,-7290,162,1458,729,-81,-81,1,19683,-65610,7290,76545,-14580,-36450,675,8100,6075,-540,-1080,10,-162,45

%N Irregular triangle read by rows: mean version of Girard-Waring formula A210258, for m = 3 data values.

%C Let SM_k = Sum( d_(t_1, t_2, t_3)* eM_1^t_1 * eM_2^t_2 * eM_3^t_3) summed over all length 3 integer partitions of k, i.e., 1*t_1+2*t_2+3*t_3=k, where SM_k are the averaged k-th power sum symmetric polynomials in 3 data (i.e., SM_k = S_k/3 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(3,k) with e_k being the k-th elementary symmetric polynomials.  The data d_(t_1, t_2, t_3) 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 The sum of the positive terms in successive rows appears to be A195350; row sums of negative terms is always 1 less than corresponding sum of positive terms.

%H Gregory Gerard Wojnar, <a href="/A287768/a287768.java.txt">Java program</a>

%H 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=3, p.22, arXiv:1706.08381 [math.GM], 2017.

%e Triangle begins:

%e     1;

%e     3,   -2;

%e     9,   -9,  1;

%e    27,  -36,  4,   6;

%e    81, -135, 15,  45,  -5;

%e   243, -486, 54, 243, -36, -18, 1;

%e   ...

%e The first few rows describe:

%e Row 1: SM_1 = 1 eM_1;

%e Row 2: SM_2 = 3*(eM_1)^2 - 2*eM_2;

%e Row 3: SM_3 = 9*(eM_1)^3 - 9*eM_1*eM_2 + 1*eM_3;

%e Row 4: SM_4 = 27*(eM_1)^4 - 36*(eM_1)^2*eM_2 + 4*eM_1*eM_3 + 6*(eM_2)^2;

%e Row 5: SM_5 = 81*(eM_1)^5 - 135*(eM_1)^3*eM_2 + 15*(eM_1)^2*eM_3 + 45*eM_1*(eM_2)^2 - 5*eM_2*eM_3.

%o (Java) // See Wojnar link.

%Y Row sums of the positive terms appears to be A195350.

%Y First entries of row n is A000244(n).

%Y Second entries of row n, for n>1, is given by -n*3^(n-2).

%Y Third entries of row n, for n>2, is given by n*3^(n-4), A006234.

%Y Fourth entries of row n, for n>3, is given by n*(n-3)*3^(n-3)/2!.

%Y Fifth entries of row n, for n>4, is given by -n*(n-4)*3^(n-5)/1!.

%Y Corresponding sequences for different sized data multisets are:  A028297 (m=2), A288199 (m=4), A288207 (m=5), A288211 (m=6), A288245 (m=7), A288188 (m=8).

%Y Cf. A210258.

%K sign,tabf

%O 1,2

%A _Gregory Gerard Wojnar_, May 31 2017