A288199 Irregular triangle read by rows: mean version of Girard-Waring formula (cf. A210258), for m = 4 data values.

1, 4, -3, 16, -18, 3, 64, -96, 16, 18, -1, 256, -480, 80, 180, -30, -5, 1024, -2304, 384, 1296, -288, -108, -24, 9, 12, 4096, -10752, 1792, 8064, -2016, -1512, 112, 252, -112, 84, -7

Offset

1,2

Comments

Let SM_k = Sum( d_(t_1, t_2, t_3, t_4)* eM_1^t_1 * eM_2^t_2 * eM_3^t_3*eM_4^t_4) summed over all length 4 integer partitions of k, i.e., 1*t_1 + 2*t_2 + 3*t_3 + 4*t_4 = k, where SM_k are the averaged k-th power sum symmetric polynomials in 4 data (i.e., SM_k = S_k/4 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(4,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2, t_3, t_4) 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.

Row sums of positive entries give 1, 4, 19, 98, 516, 2725, 14400.

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..37.

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=4, p.23, arXiv:1706.08381 [math.GM], 2017.

Example

Triangle begins:
    1;
    4,   -3;
   16,  -18,   3;
   64,  -96,  16,  18, -1;
  256, -480,  80, 180, -5, -30;
  ...
The first few rows describe:
Row 1: SM_1 = 1 eM_1;
Row 2: SM_2 = 4*(eM_1)^2 - 3*eM_2;
Row 3: SM_3 = 16*(eM_1)^3 - 18*eM_1*eM_2 + 3*eM_3;
Row 4: SM_4 = 64*(eM_1)^4 - 96*(eM_1)^2*eM_2 + 16*eM_1*eM_3 + 18*(eM_2)^2 - 1*eM_4;
Row 5: SM_5 = 256*(eM_1)^5 - 480*(eM_1)^3*eM_2 + 80*(eM_1)^2*eM_2 + 180*eM_1*(eM_2)^2 - 30*eM_2*eM_3 - 5*eM_1*eM_4.

Crossrefs

Cf. A210258, A028297 (m=2), A287768 (m=3), A288207 (m=5), A288211 (m=6), A288245 (m=7), A288188 (m=8).

Sequence in context: A280976 A285646 A092398 * A127675 A058557 A287978

Adjacent sequences: A288196 A288197 A288198 * A288200 A288201 A288202

Keyword

sign,tabf,more

Author

Gregory Gerard Wojnar, May 31 2017

Status

approved