A288211 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=6 data values.

1, 6, -5, 36, -45, 10, 216, -360, 80, 75, -10, 1296, -2700, 600, 1125, -250, -75, 5, 7776, -19440, 4320, 12150, -3600, -1125, -540, 225, 200, 36, -1, 46656, -136080, 30240, 113400, -37800, -23625, 2800, 5250, -3780, 3150, -350, 252, -105, -7

Offset: 1

Author

Gregory Gerard Wojnar, Jun 06 2017

Keywords

• sign
• tabf

Comments

Let SM_k = Sum( d_(t_1, t_2, ..., t_6)* eM_1^t_1 * eM_2^t_2 * ... * eM_6^t_6) summed over all length 6 integer partitions of k, i.e., 1*t_1 + 2*t_2 + 3*t_3 + ... + 6*t_6 = k, where SM_k are the averaged k-th power sum symmetric polynomials in 6 data (i.e., SM_k = S_k/6 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(6,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2, t_3, ..., t_6) 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 1,6,46,371,3026,24707,201748. Row sums of negative entries are always 1 less than corresponding row sums of positive entries.

Examples

			Triangle begins:
 1;
 6,-5;
 36,-45,10;
 216,-360,80,75,-10;
 1296,-2700,600,1125,-250,-75,5;
 7776,-19440,4320,12150,-3600,-1125,-540,225,200,36,-1;
 ...
Above represents:
 SM_1 = eM_1;
 SM_2 = 6*(eM_1)^2 - 5*eM_2;
 SM_3 = 36*(eM_1)^3 - 45*eM_1*eM_2 + 10*eM_3;
 SM_4 = 216*(eM_1)^4 - 360*(eM_1)^2*eM_2 + 80*eM_1*eM_3 + 75*(eM_2)^2 - 10*eM_4;
 SM_5 = 1296*(eM_1)^5 - 2700*(eM_1)^3*eM_2 + 600*(eM_1)^2*eM_3 + 1125*eM_1*(eM_2)^2 - 250*eM_2*eM_3 - 75*eM_1*eM_4 + 5*eM_5;
 ...
		

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=6, p.24, arXiv:1706.08381 [math.GM], 2017.

Crossrefs

Cf. A028297 (m=2), A287768 (m=3), A288199 (m=4), A288207 (m=5), A288245 (m=7), A288188 (m=8). Also see A210258 Girard-Waring.

First column of triangle is powers of m=6, A000400.

Programs

Java

// See link.