A288207 Irregular triangle read by rows: mean version of Girard-Waring formula A210258, for m = 5 data values.

1, 5, -4, 25, -30, 6, 125, -200, 40, 40, -4, 625, -1250, 250, 500, -25, -100, 1, 3125, -7500, 1500, 4500, -150, -1200, 6, -400, 60, 60, 15625, -43750, 8750, 35000, -875, -10500, 35, -7000, 700, 700, 1400, -14, -70

Offset

1,2

Comments

Let SM_k = Sum( d_(t_1, t_2, ... , t_5)* eM_1^t_1 * eM_2^t_2 *...* eM_5^t_5) summed over all length 5 integer partitions of k, i.e., 1*t_1+2*t_2+...+5*t_5=k, where SM_k are the averaged k-th power sum symmetric polynomials in 5 data (i.e., SM_k = S_k/5 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(5,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2,... , t_5) form a triangle, with one row for each k value starting with k=1; the number of terms in successive rows is nondecreasing.

Links

Table of n, a(n) for n=1..41.

Gregory Gerard Wojnar, Java program

Example

Triangle begins:
    1;
    5,    -4;
   25,   -30,   6;
  125,  -200,  40,  40,   -4;
  625, -1250, 250, 500, -100, -25, 1;
  ...
Above represents:
SM_1 = 1*eM_1;
SM_2 = 5*(eM_1)^2 -4*eM_2;
SM_3 = 25*(eM_1)^3 - 30*eM_1*eM_2 + 6*eM_3;
SM_4 = 125*(eM_1)^4 - 200*(eM_1)^2*eM_2 + 40*eM_1*eM_3 + 40*(eM_2)^2 - 4*eM_4;
SM_5 = 625*(eM_1)^5 - 1250*(eM_1)^3*eM_2 + 250*(eM_1)^2*eM_3 + 500*eM_1*(eM_2)^2 - 100*eM_2*eM_3 - 25*eM_1*eM_4 + 1*eM_5;
...

Prog

(Java) // See Java program link.

Crossrefs

Cf. A028297 (m=2), A287768 (m=3), A288199 (m=4), A288211 (m=6), A288245 (m=7), A288188 (m=8); A210258 Girard-Waring.

First column of triangle are powers of m=5, A000351.

Sequence in context: A341102 A167636 A180137 * A038246 A176738 A301824

Adjacent sequences: A288204 A288205 A288206 * A288208 A288209 A288210

Keyword

sign,tabf

Author

Gregory Gerard Wojnar, Jun 06 2017

Status

approved