A307449 Irregular triangle read by rows: T(n, k) gives the coefficients of the Girard-Waring formula for the sum of n-th power of five 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, 5, 1, -6, 9, 6, -2, -12, -6, 3, 6, 6, 1, -7, 14, 7, -7, -21, -7, 7, 7, 14, 7, -7, -7, 1, -8, 20, 8, -16, -32, -8, 2, 24, 12, 24, 8, -8, -8, -16, -16, 4, 8, 1, -9, 27, 9, -30, -45, -9, 9, 54, 18, 36, 9, -9, -27, -27, -27, -27, 3, 18, 9, 9, 18, -9, 1, -10, 35, 10, -50, -60, -10, 25, 100, 25, 50, 10, -2, -40, -60, -60, -40, -40, 15, 10, 10, 60, 30, 15, 30, -10, -10, -20, -20, 5

Offset: 1

Author

Wolfdieter Lang, May 14 2019

Keywords

• sign
• tabf

Comments

The length of row n is A001401(n), n >= 1.

The Girard-Waring formula for the power sum p(5,n) = Sum_{j=1..5} (x_j)^n in terms of the elementary symmetric functions e_j(x_1, x_2, x_3, x_4), for j = 1, 2 ,..., 5 is given in the W. Lang reference, Theorem 1, in an explicitly nested four sums version. See also the summary link, for N = 5 (there sigma_j^{(N)} -> e_j here).

In this array the partitions of n, with all partitions with a part >= 6 omitted, are used. Here the partitions appear in the reverse Abramowitz-Stegun order. See row n of the array of Waring numbers A115131, read backwards, with the entries corresponding to these omitted partitions.

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 19 20 21 22 23
-----------------------------------------------------------------------------
1:  1
2:  1  -2
3:  1  -3  3
4:  1  -4  2  4  -4
5:  1  -5  5  5  -5  -5  5
6:  1  -6  9  6  -2 -12 -6 3  6  6
7:  1  -7 14  7  -7 -21 -7 7  7 14  7 -7 -7
8:  1  -8 20  8 -16 -32 -8 2 24 12 24  8 -8  -8 -16 -16   4  8
9:  1  -9 27  9 -30 -45 -9 9 54 18 36  9 -9 -27 -27 -27 -27  3 18  9  9 18 -9
.
.
.
n = 10: 1 -10 35 10 -50 -60 -10 25 100 25 50 10 -2 -40 -60 -60 -40 -40 15 10 10 60 30 15 30 -10 -10 -20 -20 5.
...
------------------------------------------------------------------------------
Row n = 6: x_1^6 + x_2^6 + x_3^6 + x_4^6 + x_5^6 =  1*e_1^6  - 6*e_1^4*e_2 + 9*e_1^2*e_2^2 + 6*e_1^3*e_3 - 2*e_2^3 - 12*e_1*e_2*e_3 - 6*e_1^2*e_4 + 3*e_3^2 + 6*e_2*e_4 + 6*e_1*e_5,  with e_1 = Sum_{j=1..5} x_j, e_2 = x1*(x_2 + x_3 + x_4 + x_5) + x_2*(x_3 + x_4 + x_5) + x_3*(x_4 + x_5) + x_4*x_5, e_3 = x_1*x_2*x_3 + x_1*x_2*x_4 +  x_1*x_2*x_5 +  x_2*x_3*x_4 + x_2*x_3*x_5 + x_2*x_4*x_5 + x_3*x_4*x_5, e_4 =  x_1*x_2*x_3*x_4 + x_1*x_2*x_3*x_5 + x_1*x_2*x_4*x_5 + x_1*x_3*x_4*x_5 + x_2*x_3*x_4*x_5, e_5 = Product_{i=1..5} x_j.
		

Links

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]
• Wolfdieter Lang, On sums of powers of zeros of polynomials, J. Comp. Appl. Math. 89 (1998) 237-256; Theorem 1.
• Wolfdieter Lang, Nested sum version of the Girard-Waring formula (a summary)

Crossrefs

Cf. A001401, A115131, A132460 (N=2), A325477 (N=3), 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 >= 6.