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A115131
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Waring numbers for power sums functions in terms of elementary symmetric functions; irregular triangle T(n,k), read by rows, for n >= 1 and 1 <= k <= A000041(n).
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16
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1, -2, 1, 3, -3, 1, -4, 4, 2, -4, 1, 5, -5, -5, 5, 5, -5, 1, -6, 6, 6, 3, -6, -12, -2, 6, 9, -6, 1, 7, -7, -7, -7, 7, 14, 7, 7, -7, -21, -7, 7, 14, -7, 1, -8, 8, 8, 8, 4, -8, -16, -16, -8, -8, 8, 24, 12, 24, 2, -8, -32, -16, 8, 20, -8, 1, 9, -9, -9, -9, -9, 9, 18, 18, 9, 9, 18, 3, -9, -27, -27, -27, -27, -9, 9, 36, 18, 54, 9, -9, -45, -30, 9, 27, -9, 1
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OFFSET
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1,2
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COMMENTS
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N*t^{(N)}_n(sigma_1, ..., sigma_N):= sum((x_k)^n, k=1..N) with the elementary symmetric function sigma_k (superscript (N) omitted) in terms of the indeterminates x_1,...,x_N, is an N-variable generalization of Chebyshev's polynomials C_n((sigma_1)/2) = t^{(N=2)}_n(sigma_1, sigma_2 = 1). In general, C_n^{(N)}(sigma_1, ..., sigma_{N-1}) := t^{(N)}_n(sigma_1, ..., sigma_{N-1}, sigma_N:=1). If n > N, one puts sigma_{N+1} = 0, ..., sigma_n = 0.
The sequence of row lengths of this array is A000041(n) (partition numbers).
In row n, this triangular array uses partitions of n listed in the Abramowitz-Stegun order (compare with the M_0, M_1, M_2 and M_3 numbers given in A048996 = |A111786|, A036038, A036039 and A036040, resp.).
Row sums give (-1)^(n-1). Unsigned row sums give A000225(n)= 2^n - 1.
N*t^{(N)}_n(sigma_1, ..., sigma_N) gives the sum of the n-th power of the indeterminates x_1, ... , x_N in terms of the elementary symmetric functions of these indeterminates. The coefficient T(n, k) of this partition array corresponds to the k-th partition of n in the Abramowitz-Stegun order, and it multiplies the product of sigma_j functions encoded by this partition. See the example for n = 4 below. - Wolfdieter Lang, Mar 09 2015
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REFERENCES
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P. A. MacMahon, Combinatory Analysis, 2 vols., Chelsea, NY, 1960, see p. 5 (with a_k -> sigma_k).
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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; see pp. 831-832. [alternative scanned copy].
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FORMULA
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T(n,k) = (n/m(n,k))*A111786(n,k) for the k-th partition of n with m(n,k) parts in the Abramowitz-Stegun order for n >= 1 and k = 1..p(n), where p(n) := A000041(n).
Explicitly: T(n,k) = (-1)^(n + m(n,k)) * n * (m(n,k) - 1)!/(Product_{j = 1..n} e(k,j)!), where m(n,k):= Sum_{j = 1..n} e(k,j), with [1^e(k, 1), 2^e(k,2), ..., n^e(k,n)] being the k-th partition of n in the mentioned order. For m(n,k), see A036043.
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EXAMPLE
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First few rows of triangle T(n,k) are as follows (see the link for rows 1..10):
1;
-2, 1;
3, -3, 1;
-4, 4, 2, -4, 1;
5, -5, -5, 5, 5, -5, 1;
...
n=4: N*t^{(N)}_4 = -4*(sigma_4)^1 + 4*(sigma_1)*(sigma_3) + 2*(sigma_2)^2 -4*(sigma_1)^2*(sigma_2) + 1*(sigma_1)^4.
(For 2 <= N < 4, one puts sigma_{N+1} = 0 = ... = sigma_4 = 0.) This becomes Sum_{k = 1..N} (x_k)^4 if the sigma functions are written in terms of the variables x_1, x_2, ..., x_N. E.g., for N=2: 0 + 0 + 2*(x_1*x_2)^2 -4*(x_1 + x_2)^2*(x_1*x_2) + 1*(x_1 + x_2)^4 = (x_1)^4 + (x_2)^4.
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CROSSREFS
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KEYWORD
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sign,easy,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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