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A287768
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Irregular triangle read by rows: mean version of Girard-Waring formula A210258, for m = 3 data values.
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12
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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Triangle begins:
1;
3, -2;
9, -9, 1;
27, -36, 4, 6;
81, -135, 15, 45, -5;
243, -486, 54, 243, -36, -18, 1;
...
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.
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PROG
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(Java) See link
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CROSSREFS
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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!.
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KEYWORD
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sign,tabf
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AUTHOR
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STATUS
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approved
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