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 A288211 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=6 data values. 7
 1, 6, -5, 36, -45, 10, 216, -360, 80, 75, -10, 1296, -2700, 600, 1125, -250, -75, 5, 7776, -19440, 4320, 12150, -3600, -1125, -540, 225, 200, 36, -1, 46656, -136080, 30240, 113400, -37800, -23625, 2800, 5250, -3780, 3150, -350, 252, -105, -7 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Let SM_k = Sum( d_(t_1, t_2, ..., t_6)* eM_1^t_1 * eM_2^t_2 * ... * eM_6^t_6) summed over all length 6 integer partitions of k, i.e., 1*t_1 + 2*t_2 + 3*t_3 + ... + 6*t_6 = k, where SM_k are the averaged k-th power sum symmetric polynomials in 6 data (i.e., SM_k = S_k/6 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(6,k) with e_k being the k-th elementary symmetric polynomials.  The data d_(t_1, t_2, t_3, ..., t_6) 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 1,6,46,371,3026,24707,201748. 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, Table GW.n=6, p.24, arXiv:1706.08381 [math.GM], 2017. EXAMPLE Triangle begins: 1; 6,-5; 36,-45,10; 216,-360,80,75,-10; 1296,-2700,600,1125,-250,-75,5; 7776,-19440,4320,12150,-3600,-1125,-540,225,200,36,-1; ... Above represents: SM_1 = eM_1; SM_2 = 6*(eM_1)^2 - 5*eM_2; SM_3 = 36*(eM_1)^3 - 45*eM_1*eM_2 + 10*eM_3; SM_4 = 216*(eM_1)^4 - 360*(eM_1)^2*eM_2 + 80*eM_1*eM_3 + 75*(eM_2)^2 - 10*eM_4; SM_5 = 1296*(eM_1)^5 - 2700*(eM_1)^3*eM_2 + 600*(eM_1)^2*eM_3 + 1125*eM_1*(eM_2)^2 - 250*eM_2*eM_3 - 75*eM_1*eM_4 + 5*eM_5; ... PROG See Java program link. CROSSREFS Cf. A028297 (m=2), A287768 (m=3), A288199 (m=4), A288207 (m=5), A288245 (m=7), A288188 (m=8). Also see A210258 Girard-Waring. First column of triangle is powers of m=6, A000400. Sequence in context: A137763 A029763 A283980 * A038259 A302750 A268000 Adjacent sequences:  A288208 A288209 A288210 * A288212 A288213 A288214 KEYWORD sign,tabf AUTHOR Gregory Gerard Wojnar, Jun 06 2017 STATUS approved

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Last modified June 15 13:33 EDT 2021. Contains 345048 sequences. (Running on oeis4.)