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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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%I #64 Dec 15 2019 16:26:08

%S 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,

%T 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,

%U -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

%N 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).

%C 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.

%C The sequence of row lengths of this array is A000041(n) (partition numbers).

%C 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.).

%C Row sums give (-1)^(n-1). Unsigned row sums give A000225(n)= 2^n - 1.

%C 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

%D P. A. MacMahon, Combinatory Analysis, 2 vols., Chelsea, NY, 1960, see p. 5 (with a_k -> sigma_k).

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972; see pp. 831-832. [alternative scanned copy].

%H Wolfdieter Lang, <a href="http://dx.doi.org/10.1016/S0377-0427(97)00240-9">On sums of powers of zeros of polynomials</a>, J. Comp. Appl. Math. 89 (1998) 237-256; see Theorem 1.

%H Wolfdieter Lang, <a href="/A115131/a115131.pdf">First 10 rows of the array</a>.

%H R. Lidl, <a href="http://dx.doi.org/10.1515/crll.1975.273.178">Tschebyscheffpolynome in mehreren Variablen</a>, J. reine u. angew. Math. 273 (1975), 178-198.

%H R. Lidl, <a href="https://eudml.org/doc/151555">Tschebyscheffpolynome in mehreren Variablen</a>, J. reine u. angew. Math. 273 (1975), 178-198.

%H R. Lidl and Ch. Wells, <a href="http://dx.doi.org/10.1515/crll.1972.255.104">Chebyshev polynomials in several variables</a>, J. reine u. angew. Math. 255 (1972), 104-111.

%H R. Lidl and Ch. Wells, <a href="https://eudml.org/doc/151238">Chebyshev polynomials in several variables</a>, J. reine u. angew. Math. 255 (1972), 104-111.

%H P. A. MacMahon, <a href="http://name.umdl.umich.edu/ABU9009.0001.001">Combinatory analysis</a> (2 vols.), Chelsea, NY, 1960; see p. 5 (with a_k -> sigma_k).

%F 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).

%F 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.

%e First few rows of triangle T(n,k) are as follows (see the link for rows 1..10):

%e 1;

%e -2, 1;

%e 3, -3, 1;

%e -4, 4, 2, -4, 1;

%e 5, -5, -5, 5, 5, -5, 1;

%e ...

%e 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.

%e (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.

%Y Cf. A000041, A000225, A036038, A036039, A036040, A048996, A111786.

%Y Cf. A210258 (in another ordering of partitions), A132460 (N=2), A325477 (N=3),

%Y A324602 (N=4).

%K sign,easy,tabf

%O 1,2

%A _Wolfdieter Lang_, Jan 13 2006

%E Various sections edited by _Petros Hadjicostas_, Dec 14 2019