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 A288245 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=7 data values. 7
 1, 7, -6, 49, -63, 15, 343, -588, 140, 126, -20, 2401, -5145, 1225, 2205, -175, -525, 15, 16807, -43218, 10290, 27783, -1470, -8820, 126, -2646, 630, 525, -6, 117649, -352947, 84035, 302526, -12005, -108045, 1029, -64827, 10290, 8575, -49, 15435, -441, -1225, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Let SM_k = Sum( d_(t_1, t_2, t_3, ..., t_7)* eM_1^t_1 * eM_2^t_2 * ... * eM_7^t_7) summed over all length 7 integer partitions of k, i.e., 1*t_1 + 2*t_2 + 3*t_3 + ... + 7*t_7 = k, where SM_k are the averaged k-th power sum symmetric polynomials in 7 data (i.e., SM_k = S_k/7 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(7,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2, t_3, ..., t_7) 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 1,7,64,609,5846,56161,... 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], 2107. EXAMPLE Triangular array begins... 1; 7,-6; 49,-63,15; 343,-588,140,126,-20; 2401,-5145,1225,2205,-175,-525,15; 16807,-43218,10290,27783,-1470,-8820,126,-2646,630,525,-6; 117649,-352947,84035,302526,-12005,-108045,1029,64827,10290,8575,-49,15435,-441,-1225,1; PROG (Java) see links CROSSREFS Cf. A028297 (m=2), A287768 (m=3), A288199 (m=4), A288211 (m=5), A288211 (m=6), A288188 (m=8). Also see Girard-Waring A210258. First entries of each row of triangle are powers of m=7, A000420. Sequence in context: A070425 A163842 A328697 * A038272 A249992 A223531 Adjacent sequences:  A288242 A288243 A288244 * A288246 A288247 A288248 KEYWORD sign,tabf AUTHOR Gregory Gerard Wojnar, Jun 06 2017 STATUS approved

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Last modified September 17 22:14 EDT 2021. Contains 347489 sequences. (Running on oeis4.)