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A288245 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=7 data values. 6
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

Table of n, a(n) for n=1..44.

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: A073112 A070425 A163842 * 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 February 18 23:16 EST 2018. Contains 299330 sequences. (Running on oeis4.)