

A288188


Irregular triangle read by rows of normalized GirardWaring 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
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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 kth power sum symmetric polynomials in 8 data (i.e., SM_k = S_k/8 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(8,k) with e_k being the kth 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

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], 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 GirardWaring A210258. T(n,1)=8^(n1)=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



