1,2

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 T_n((sigma_1)/2) = t^{(N=2)}_n(sigma_1,sigma_2=1) polynomials. In general T_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).

This array uses in row number n partitions of n listed in the Abramowitz-Stegun order (compare with the M_0, M_1, M_2 and M_3 numbers given in |A111786|, A036038, A036039 and A036040, resp.).

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

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

R. Lidl, Ch. Wells, 'Chebyshev polynomials in several variables', J. reine u. angew. Math. 255 (1972)104-111.

R. Lidl, `Tschebyscheffpolynome in mehreren Variablen`, J. reine u. angew. Math. 273 (1975)178-198.

W. Lang: 'On Sums of Powers of Zeros of Polynomials', J. of Comp. and Applied Math. 89 (1998) 237-256; Theorem 1.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].

P. A. MacMahon, Combinatory analysis.

W. Lang: First 10 rows.

a(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. n>=1, k=1, ..., p(n):=A000041(n).

Explicitly: a(n, k)= n*(m(n, k)-1)!/product(e(k, j)!, j=1..n ), where m(n, k):= sum(e(k, j), j=1..n), with [1^e(k, 1), 2^e(k, 2), ..., n^e(k, n)] the k-th partition of n in the mentioned order. For m(n, k) see A036043.

[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((x_k)^4,k=1..N) if the

sigma functions are written in terms of the x_1,...,x_N variables. E.g. N=2:

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.

Sequence in context: A049834 A134625 A054531 this_sequence A117895 A128139 A104732

Adjacent sequences: A115128 A115129 A115130 this_sequence A115132 A115133 A115134

sign,easy,tabf

Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Jan 13 2006