A325477 Irregular triangle read by rows: T(n, k) gives the coefficients of the Girard-Waring formula for the sum of n-th power of three indeterminates in terms of their elementary symmetric functions.

1, 1, -2, 1, -3, 3, 1, -4, 2, 4, 1, -5, 5, 5, -5, 1, 1, -6, 9, 6, -2, -12, 3, 1, -7, 14, 7, -7, -21, 7, 7, 1, -8, 20, 8, -16, -32, 2, 24, 12, -8, 1, -9, 27, 9, -30, -45, 9, 54, 18, -9, -27, 3, 1, -10, 35, 10, -50, -60, 25, 100, 25, -2, -40, -60, 15, 10

Offset: 1

Author

Wolfdieter Lang, May 03 2019

Keywords

• sign
• tabf
• easy

Comments

The length of row n is A001399(n), n >= 1.

The Girard-Waring formula for the power sum p(3,n) = x1^n + x2^2 + x3^n in terms of the elementary symmetric functions e_j(x1, x2, x3), for j=1, 2, 3, is given by Sum_{i=0..floor(n/3)} Sum_{j=0...floor((n-3*i)/2)} ((-1)^j)*n*(n - j - 2*i - 1)!/(i!*j!*(n - 2*j -3*i)!)*e_1^(n-3*i-2*j)*(e_2)^j*(e_3)^i, n >= 1 (the arguments of e_j have been omitted). See the W. Lang reference, Theorem 1, case N = 3, with r -> n.

This is an array using the partitions of n, in the reverse Abramowitz-Stegun order, with all partitions which have a part larger than 3 elininated. See row n of the array of Waring numbers A115131 read backwards, with these partitions omitted, and numerated with k from 1, 2, ..., A001399(n).

Examples

			The irregular triangle T(n, k) begins:
n\k  1   2  3   4   5   6  7   8   9  10  11  12  13  14 ...
-------------------------------------------------------------
1:   1
2:   1  -2
3:   1  -3  3
4:   1  -4  2   4
5:   1  -5  5   5  -5
6:   1  -6  9   6  -2 -12  3
7:   1  -7 14   7  -7 -21  7   7
8:   1  -8 20   8 -16 -32  2  24  12  -8
9:   1  -9 27   9 -30 -45  9  54  18  -9 -27   3
10:  1 -10 35  10 -50 -60 25 100  25  -2 -40 -60  15  10
...
n = 4: x1^4 + x2^4 + x3^4 = (e_1)^4 - 4*(e_1)^2*e_2 + 2*(e_2)^2 + 4*e_1*e_3, with e_1 = x1 + x2 + x3, e_2 = x1*x2 + x1*x3 + x2*x^3 and e_3 = x1*x2*x3.
		

Links

• Wolfdieter Lang, On sums of powers of zeros of polynomials, J. Comp. Appl. Math. 89 (1998) 237-256.

Crossrefs

Cf. A001399, A115131, A132460 (case N=2), A324602 (N=4).

Formula

T(n, k) is the k-th coefficient of the Waring number partition array A115131(n, m) (k there is replaced here by m), read backwards, omitting all partitions which have a part >= 3.