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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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Table of n, a(n) for n=1..37.
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=4, p.23, arXiv:1706.08381 [math.GM], 2017.
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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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Cf. A210258, A028297 (m=2), A287768 (m=3), A288207 (m=5), A288211 (m=6), A288245 (m=7), A288188 (m=8).
Sequence in context: A280976 A285646 A092398 * A127675 A058557 A287978
Adjacent sequences: A288196 A288197 A288198 * A288200 A288201 A288202
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KEYWORD
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sign,tabf,more
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AUTHOR
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Gregory Gerard Wojnar, May 31 2017
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STATUS
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approved
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