A325477 -id:A325477 - cp's OEIS frontend

A115131 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).

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

Offset: 1

Wolfdieter Lang, Jan 13 2006

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

			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.

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

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].
Wolfdieter Lang, On sums of powers of zeros of polynomials, J. Comp. Appl. Math. 89 (1998) 237-256; see Theorem 1.
Wolfdieter Lang, First 10 rows of the array.
R. Lidl, Tschebyscheffpolynome in mehreren Variablen, J. reine u. angew. Math. 273 (1975), 178-198.
R. Lidl, Tschebyscheffpolynome in mehreren Variablen, J. reine u. angew. Math. 273 (1975), 178-198.
R. Lidl and Ch. Wells, Chebyshev polynomials in several variables, J. reine u. angew. Math. 255 (1972), 104-111.
R. Lidl and Ch. Wells, Chebyshev polynomials in several variables, J. reine u. angew. Math. 255 (1972), 104-111.
P. A. MacMahon, Combinatory analysis (2 vols.), Chelsea, NY, 1960; see p. 5 (with a_k -> sigma_k).

Cf. A000041, A000225, A036038, A036039, A036040, A048996, A111786.
Cf. A210258 (in another ordering of partitions), A132460 (N=2), A325477 (N=3),
A324602 (N=4).

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.

Various sections edited by Petros Hadjicostas, Dec 14 2019

A324602 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).

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, May 03 2019

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.

			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.

Wolfdieter Lang, On sums of powers of zeros of polynomials, J. Comp. Appl. Math. 89 (1998) 237-356.

Cf. A001400, A115131, A132460 (N=2), A325477 (N=3).

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.

A307449 Irregular triangle read by rows: T(n, k) gives the coefficients of the Girard-Waring formula for the sum of n-th power of five indeterminates in terms of their elementary symmetric functions (reverse Abramowitz-Stegun order of partitions).

Original entry on oeis.org

1, 1, -2, 1, -3, 3, 1, -4, 2, 4, -4, 1, -5, 5, 5, -5, -5, 5, 1, -6, 9, 6, -2, -12, -6, 3, 6, 6, 1, -7, 14, 7, -7, -21, -7, 7, 7, 14, 7, -7, -7, 1, -8, 20, 8, -16, -32, -8, 2, 24, 12, 24, 8, -8, -8, -16, -16, 4, 8, 1, -9, 27, 9, -30, -45, -9, 9, 54, 18, 36, 9, -9, -27, -27, -27, -27, 3, 18, 9, 9, 18, -9, 1, -10, 35, 10, -50, -60, -10, 25, 100, 25, 50, 10, -2, -40, -60, -60, -40, -40, 15, 10, 10, 60, 30, 15, 30, -10, -10, -20, -20, 5

Offset: 1

Wolfdieter Lang, May 14 2019

The length of row n is A001401(n), n >= 1.
The Girard-Waring formula for the power sum p(5,n) = Sum_{j=1..5} (x_j)^n in terms of the elementary symmetric functions e_j(x_1, x_2, x_3, x_4), for j = 1, 2 ,..., 5 is given in the W. Lang reference, Theorem 1, in an explicitly nested four sums version. See also the summary link, for N = 5 (there sigma_j^{(N)} -> e_j here).
In this array the partitions of n, with all partitions with a part >= 6 omitted, are used. Here the partitions appear in the reverse Abramowitz-Stegun order. See row n of the array of Waring numbers A115131, read backwards, with the entries corresponding to these omitted partitions.

			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 19 20 21 22 23
-----------------------------------------------------------------------------
1:  1
2:  1  -2
3:  1  -3  3
4:  1  -4  2  4  -4
5:  1  -5  5  5  -5  -5  5
6:  1  -6  9  6  -2 -12 -6 3  6  6
7:  1  -7 14  7  -7 -21 -7 7  7 14  7 -7 -7
8:  1  -8 20  8 -16 -32 -8 2 24 12 24  8 -8  -8 -16 -16   4  8
9:  1  -9 27  9 -30 -45 -9 9 54 18 36  9 -9 -27 -27 -27 -27  3 18  9  9 18 -9
.
.
.
n = 10: 1 -10 35 10 -50 -60 -10 25 100 25 50 10 -2 -40 -60 -60 -40 -40 15 10 10 60 30 15 30 -10 -10 -20 -20 5.
...
------------------------------------------------------------------------------
Row n = 6: x_1^6 + x_2^6 + x_3^6 + x_4^6 + x_5^6 =  1*e_1^6  - 6*e_1^4*e_2 + 9*e_1^2*e_2^2 + 6*e_1^3*e_3 - 2*e_2^3 - 12*e_1*e_2*e_3 - 6*e_1^2*e_4 + 3*e_3^2 + 6*e_2*e_4 + 6*e_1*e_5,  with e_1 = Sum_{j=1..5} x_j, e_2 = x1*(x_2 + x_3 + x_4 + x_5) + x_2*(x_3 + x_4 + x_5) + x_3*(x_4 + x_5) + x_4*x_5, e_3 = x_1*x_2*x_3 + x_1*x_2*x_4 +  x_1*x_2*x_5 +  x_2*x_3*x_4 + x_2*x_3*x_5 + x_2*x_4*x_5 + x_3*x_4*x_5, e_4 =  x_1*x_2*x_3*x_4 + x_1*x_2*x_3*x_5 + x_1*x_2*x_4*x_5 + x_1*x_3*x_4*x_5 + x_2*x_3*x_4*x_5, e_5 = Product_{i=1..5} x_j.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]
Wolfdieter Lang, On sums of powers of zeros of polynomials, J. Comp. Appl. Math. 89 (1998) 237-256; Theorem 1.
Wolfdieter Lang, Nested sum version of the Girard-Waring formula (a summary)

Cf. A001401, A115131, A132460 (N=2), A325477 (N=3), A324602 (N=4).

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 >= 6.

cp's OEIS Frontend

A115131 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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A324602 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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A307449 Irregular triangle read by rows: T(n, k) gives the coefficients of the Girard-Waring formula for the sum of n-th power of five indeterminates in terms of their elementary symmetric functions (reverse Abramowitz-Stegun order of partitions).

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Comments

Examples

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