A324602 Irregular triangle read by rows: T(n, k) gives the coefficients of the Girard-Waring formula for the sum of n-th power of four indeterminates in terms of their elementary symmetric functions (reverse Abramowitz-Stegun order of partitions).

1, 1, -2, 1, -3, 3, 1, -4, 2, 4, -4, 1, -5, 5, 5, -5, -5, 1, -6, 9, -2, 6, -12, 3, -6, 6, 1, -7, 14, -7, 7, -21, 7, 7, -7, 14, -7, 1, -8, 20, -16, 2, 8, -32, 24, 12, -8, -8, 24, -8, -16, 4, 1, -9, 27, -30, 9, 9, -45, 54, -9, 18, -27, 3, -9, 36, -27, -27, 18, 9, 1, -10, 35, -50, 25, -2, 10, -60, 100, -40, 25, -60, 15, 10, -10, 50, -60, 10, -40, 60, -10, 15, -10

Offset: 1

Author

Wolfdieter Lang, May 03 2019

Keywords

• sign
• tabf
• easy

Comments

The length of row n is A001400(n), n >= 1.

The Girard-Waring formula for the power sum p(4,n) := Sum_{j=1..4} (x_j)^n in terms of the elementary symmetric functions e_j(x_1, x_2, x_3, x_4), for j = 1, 2, 3, 4, is given by Sum_{i1=0..floor(n/4)} Sum_{i2=0...floor((n-4*i1)/3)} Sum_{i3=0...floor((n-4*i1-3*i2)/2)} ((-1)^(i1 + i3))*n*(n-1-i3-2*i2-3*i1)!/(i1!*i2!*i3!*(n-2*i3-3*i2-4*i1)!)*e_1^(n-2*i3-3*i2-4*i1)*(e_2)^i3*(e_3)^i2*(e_4)^i1, n >= 1 (the arguments of e_j have been omitted). See the W. Lang reference, Theorem 1, case N = 4, with r -> n.

This is an array using the partitions of n, in the reverse Abramowitz-Stegun order, with all partitions with a part >= 5 eliminated. See row n of the array of Waring numbers A115131, read backwards, with these partitions omitted.

Examples

			The irregular triangle T(n, k) begins:
n\k 1   2  3  4   5   6  7  8   9  10  11  12  13   14   15  16  17 18 ...
-----------------------------------------------------------------------------
1:  1
2:  1  -2
3:  1  -3  3
4:  1  -4  2  4  -4
5:  1  -5  5  5  -5  -5
6:  1  -6  9  6  -2 -12 -6  3   6
7:  1  -7 14  7  -7 -21 -7  7   7  14  -7
8:  1  -8 20  8 -16 -32 -8  2  24  12  24  -8  -8  -16    4
9:  1  -9 27  9 -30 -45 -9  9  54  18  36  -9 -27  -27  -27   3  18  9
...
n = 10: 1 -10 35 10 -50 -60 -10 25 100 25 50 -2 -40 -60 -60 -40 15 10 10 60 15 -10 -10.
...
-----------------------------------------------------------------------------
Row n = 5: p(4,5) = x_1^5 + x_2^5 + x_3^5 + x_4^5 =  1*e_1^5  - 5* e_1^3*e_2  + 5*e_1*e_2^2 + 5*e_1^2*e_3 - 5*e_2*e_3 - 5*e_1*e_4,
  with e_1 = Sum_{j=1..4} x_j, e_2 = x1*(x_2 + x_3 + x_4) + x_2*(x_3 + x_4) + x_3*x_4, e_3 = x_1*x_2*x_3 + x_1*x_2*x_4 + x_2*x_3*x_4, e_4 = Product_{i=1..4} x_j.
		

Links

• Wolfdieter Lang, On sums of powers of zeros of polynomials, J. Comp. Appl. Math. 89 (1998) 237-356.

Crossrefs

Cf. A001400, A115131, A132460 (N=2), A325477 (N=3).

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 >= 5.