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 A287768 Irregular triangle read by rows: mean version of Girard-Waring formula A210258, for m = 3 data values. 12
 1, 3, -2, 9, -9, 1, 27, -36, 4, 6, 81, -135, 15, 45, -5, 243, -486, 54, 243, -36, -18, 1, 729, -1701, 189, 1134, -189, -189, 7, 21, 2187, -5832, 648, 4860, -864, -1296, 36, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. 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. LINKS Gregory Gerard Wojnar, java program G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, Table GW.n=3, p.22, arXiv:1706.08381 [math, GM], 2017. EXAMPLE Triangle begins:    1;    3,   -2;    9,   -9,  1;   27,  -36,  4,  6;   81, -135, 15, 45, -5;   ... The first few rows describe: Row 1: SM_1 = 1 eM_1; Row 2: SM_2 = 3*(eM_1)^2 - 2*eM_2; Row 3: SM_3 = 9*(eM_1)^3 - 9*eM_1*eM_2 + 1*eM_3; Row 4: SM_4 = 27*(eM_1)^4 - 36*(eM_1)^2*eM_2 + 4*eM_1*eM_3 + 6*(eM_2)^2; 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. PROG (Java) See link CROSSREFS Row sums of the positive terms appears to be A195350. First entries of row n is A000244(n). Second entries of row n, for n>1, is given by -n*3^(n-2). Third entries of row n, for n>2, is given by n*3^(n-4), A006234. Fourth entries of row n, for n>3, is given by n*(n-3)*3^(n-3)/2!. Fifth entries of row n, for n>4, is given by -n*(n-4)*3^(n-5)/1!. 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). Cf. A210258. Sequence in context: A050676 A010372 A199455 * A197831 A244995 A152049 Adjacent sequences:  A287765 A287766 A287767 * A287769 A287770 A287771 KEYWORD sign,tabf AUTHOR Gregory Gerard Wojnar, May 31 2017 STATUS approved

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Last modified January 16 15:53 EST 2019. Contains 319195 sequences. (Running on oeis4.)