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A325477
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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 three indeterminates in terms of their elementary symmetric functions.
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3
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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
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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 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).
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LINKS
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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 >= 3.
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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 ...
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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.
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CROSSREFS
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KEYWORD
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sign,tabf,easy
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AUTHOR
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STATUS
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approved
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