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 A288188 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=8 data values. 7
 1, 8, -7, 64, -84, 21, 512, -896, 224, 196, -35, 4096, -8960, 2240, 3920, -350, -980, 35, 32768, -86016, 21504, 56448, -3360, -18816, 336, -5488, 1470, 1176, -21, 262144, -802816, 200704, 702464, -31360, -263424, 3136, -153664, 27440, 21952, -196, 38416, -1372, -3430, 7 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Let SM_k = Sum( d_(t_1, t_2, t_3, ..., t_8)* eM_1^t_1 * eM_2^t_2 * ...*eM_8^t_8) summed over all length 8 integer partitions of k, i.e., 1*t_1+2*t_2+3*t_3+...+8*t_8=k, where SM_k are the averaged k-th power sum symmetric polynomials in 8 data (i.e., SM_k = S_k/8 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(8,k) with e_k being the k-th elementary symmetric polynomials.  The data d_(t_1, t_2, t_3, ..., t_8) form a triangle, with one row for each k value starting with k=1; the number of terms in successive rows is nondecreasing. Row sums of positive entries give: 1,8,85,932,10291,114878,... Row sums of negative entries are always 1 less than corresponding row sums of positive entries. LINKS Gregory Gerard Wojnar, Java program G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, pp. 22-24, arXiv:1706.08381 [math.GM], 2017. EXAMPLE Triangle begins      1;      8,    -7;     64,   -84,   21;    512,  -896,  224,  196,  -35;   4096, -8960, 2240, 3920, -350, -980, 35;   ... PROG (Java) See link. CROSSREFS Cf. A028297 (m=2), A287768 (m=3), A288199 (m=4), A288207 (m=5), A288211 (m=6), A288245 (m=7). See Girard-Waring A210258. T(n,1)=8^(n-1)=A001018(n). Sequence in context: A286460 A317231 A237646 * A038285 A261117 A098432 Adjacent sequences:  A288185 A288186 A288187 * A288189 A288190 A288191 KEYWORD sign,tabf AUTHOR Gregory Gerard Wojnar, Jun 16 2017 STATUS approved

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Last modified April 11 22:01 EDT 2021. Contains 342888 sequences. (Running on oeis4.)