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 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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 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. 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) 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. EXAMPLE 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. CROSSREFS Cf.  A001401, A115131, A132460 (N=2), A325477 (N=3), A324602 (N=4). Sequence in context: A054531 A324602 A319226 * A207645 A115131 A263916 Adjacent sequences:  A307446 A307447 A307448 * A307450 A307451 A307452 KEYWORD sign,tabf AUTHOR Wolfdieter Lang, May 14 2019 STATUS approved

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Last modified August 3 20:08 EDT 2020. Contains 336201 sequences. (Running on oeis4.)