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A288188 Irregular triangle read by rows of normalized Girard-Waring 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 (list; graph; refs; listen; history; text; internal format)
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 k-th power sum symmetric polynomials in 8 data (i.e., SM_k = S_k/8 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(8,k) with e_k being the k-th 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. 22-24, 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 Girard-Waring A210258. T(n,1)=8^(n-1)=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

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Last modified November 30 17:12 EST 2021. Contains 349424 sequences. (Running on oeis4.)