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#22 by Michel Marcus at Thu Jun 29 10:43:47 EDT 2017
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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=4, p.23, arXiv:1706.08381 [math.GM], 2017.
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Discussion
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| Thu Jun 29
| 10:44
| Michel Marcus: usual info for arxiv link
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#21 by Gregory Gerard Wojnar at Thu Jun 29 10:39:24 EDT 2017
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#20 by Gregory Gerard Wojnar at Thu Jun 29 10:38:51 EDT 2017
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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=4, p.23.
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| Thu Jun 29
| 10:39
| Gregory Gerard Wojnar: added arXiv paper link, with Table/page reference.
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#19 by N. J. A. Sloane at Fri Jun 16 22:35:55 EDT 2017
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#18 by Gregory Gerard Wojnar at Fri Jun 16 09:50:01 EDT 2017
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| Fri Jun 16
| 22:35
| N. J. A. Sloane: OK, I accept that suggestion! "Values" is better than "points".
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#17 by Gregory Gerard Wojnar at Fri Jun 16 09:47:29 EDT 2017
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Irregular triangle read by rows: mean version of Girard-Waring formula (cf. A210258, ), for m = 4 data pointsvalues.
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| Fri Jun 16
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| Gregory Gerard Wojnar: I added slight modifications: "values" rather than "points"-- to me "points" connotes bivariate (or multivariate) data, rather than single values in $\Cbb^1$ or $\Rbb^1$.
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#16 by N. J. A. Sloane at Thu Jun 15 17:03:00 EDT 2017
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#15 by N. J. A. Sloane at Thu Jun 15 17:02:38 EDT 2017
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Irregular triangle read by rows: Meanmean version of Girard-Waring formula A210258, for m = 4 data points.
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| Thu Jun 15
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| N. J. A. Sloane: Is my version of the definition OK?
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#14 by Jon E. Schoenfield at Sun Jun 11 16:27:48 EDT 2017
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#13 by Jon E. Schoenfield at Sun Jun 11 16:27:45 EDT 2017
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Irregular triangle read by rows. : Mean version of Girard-Waring formula for m = 4 data.
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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.
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1;
4, -, -3;
16, -, -18, 3;
64, -, -96, 16, 18, -1;
256, -480, 80, 180, -5, -30;
...
...
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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