A287768 Irregular triangle read by rows: mean version of Girard-Waring formula A210258, for m = 3 data values.

1, 3, -2, 9, -9, 1, 27, -36, 4, 6, 81, -135, 15, 45, -5, 243, -486, 54, 243, -36, -18, 1, 729, -1701, 189, 1134, -189, -189, 7, 21, 2187, -5832, 648, 4860, -864, -1296, 36, 216, 54, -8, 6561, -19683, 2187, 19683, -3645, -7290, 162, 1458, 729, -81, -81, 1, 19683, -65610, 7290, 76545, -14580, -36450, 675, 8100, 6075, -540, -1080, 10, -162, 45

Offset: 1

Author

Gregory Gerard Wojnar, May 31 2017

Keywords

• sign
• tabf

Comments

Let SM_k = Sum( d_(t_1, t_2, t_3)* eM_1^t_1 * eM_2^t_2 * eM_3^t_3) summed over all length 3 integer partitions of k, i.e., 1*t_1+2*t_2+3*t_3=k, where SM_k are the averaged k-th power sum symmetric polynomials in 3 data (i.e., SM_k = S_k/3 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(3,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2, t_3) form an irregular triangle, with one row for each k value starting with k=1; "irregular" means that the number of terms in successive rows is nondecreasing.

The sum of the positive terms in successive rows appears to be A195350; row sums of negative terms is always 1 less than corresponding sum of positive terms.

Examples

			Triangle begins:
    1;
    3,   -2;
    9,   -9,  1;
   27,  -36,  4,   6;
   81, -135, 15,  45,  -5;
  243, -486, 54, 243, -36, -18, 1;
  ...
The first few rows describe:
Row 1: SM_1 = 1 eM_1;
Row 2: SM_2 = 3*(eM_1)^2 - 2*eM_2;
Row 3: SM_3 = 9*(eM_1)^3 - 9*eM_1*eM_2 + 1*eM_3;
Row 4: SM_4 = 27*(eM_1)^4 - 36*(eM_1)^2*eM_2 + 4*eM_1*eM_3 + 6*(eM_2)^2;
Row 5: SM_5 = 81*(eM_1)^5 - 135*(eM_1)^3*eM_2 + 15*(eM_1)^2*eM_3 + 45*eM_1*(eM_2)^2 - 5*eM_2*eM_3.
		

Links

• Gregory Gerard Wojnar, Java program
• G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, Table GW.n=3, p.22, arXiv:1706.08381 [math.GM], 2017.

Crossrefs

Row sums of the positive terms appears to be A195350.

First entries of row n is A000244(n).

Second entries of row n, for n>1, is given by -n*3^(n-2).

Third entries of row n, for n>2, is given by n*3^(n-4), A006234.

Fourth entries of row n, for n>3, is given by n*(n-3)*3^(n-3)/2!.

Fifth entries of row n, for n>4, is given by -n*(n-4)*3^(n-5)/1!.

Corresponding sequences for different sized data multisets are: A028297 (m=2), A288199 (m=4), A288207 (m=5), A288211 (m=6), A288245 (m=7), A288188 (m=8).

Cf. A210258.

Programs

Java

// See Wojnar link.