

A288245


Irregular triangle read by rows of normalized GirardWaring 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 kth power sum symmetric polynomials in 7 data (i.e., SM_k = S_k/7 where S_k are the kth power sum symmetric polynomials, and where eM_k are the averaged kth elementary symmetric polynomials, eM_k = e_k/binomial(7,k) with e_k being the kth 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. 2224, 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 GirardWaring 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



