A288188 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=8 data values.

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

Offset: 1

Author

Gregory Gerard Wojnar, Jun 16 2017

Keywords

• sign
• tabf

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.

Examples

			Triangle begins
     1;
     8,    -7;
    64,   -84,   21;
   512,  -896,  224,  196,  -35;
  4096, -8960, 2240, 3920, -350, -980, 35;
  ...
		

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], 2017.

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).

Programs

Java

// See Wojnar link.