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A288199
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Irregular triangle read by rows: mean version of Girard-Waring formula (cf. A210258), for m = 4 data values.
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7
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1, 4, -3, 16, -18, 3, 64, -96, 16, 18, -1, 256, -480, 80, 180, -30, -5, 1024, -2304, 384, 1296, -288, -108, -24, 9, 12, 4096, -10752, 1792, 8064, -2016, -1512, 112, 252, -112, 84, -7
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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, t_4)* eM_1^t_1 * eM_2^t_2 * eM_3^t_3*eM_4^t_4) summed over all length 4 integer partitions of k, i.e., 1*t_1 + 2*t_2 + 3*t_3 + 4*t_4 = k, where SM_k are the averaged k-th power sum symmetric polynomials in 4 data (i.e., SM_k = S_k/4 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(4,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2, t_3, t_4) 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.
Row sums of positive entries give 1, 4, 19, 98, 516, 2725, 14400.
Row sums of negative entries are always 1 less than corresponding row sums of positive entries.
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LINKS
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EXAMPLE
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Triangle begins:
1;
4, -3;
16, -18, 3;
64, -96, 16, 18, -1;
256, -480, 80, 180, -5, -30;
...
The first few rows describe:
Row 1: SM_1 = 1 eM_1;
Row 2: SM_2 = 4*(eM_1)^2 - 3*eM_2;
Row 3: SM_3 = 16*(eM_1)^3 - 18*eM_1*eM_2 + 3*eM_3;
Row 4: SM_4 = 64*(eM_1)^4 - 96*(eM_1)^2*eM_2 + 16*eM_1*eM_3 + 18*(eM_2)^2 - 1*eM_4;
Row 5: SM_5 = 256*(eM_1)^5 - 480*(eM_1)^3*eM_2 + 80*(eM_1)^2*eM_2 + 180*eM_1*(eM_2)^2 - 30*eM_2*eM_3 - 5*eM_1*eM_4.
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CROSSREFS
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KEYWORD
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sign,tabf,more
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AUTHOR
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STATUS
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approved
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