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A307449
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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).
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1
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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
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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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]
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FORMULA
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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.
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EXAMPLE
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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
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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
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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.
...
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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.
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CROSSREFS
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KEYWORD
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sign,tabf
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AUTHOR
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STATUS
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approved
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