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A324602
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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 four indeterminates in terms of their elementary symmetric functions (reverse Abramowitz-Stegun order of partitions).
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5
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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
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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 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.
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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 >= 5.
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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 ...
-----------------------------------------------------------------------------
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.
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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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