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A111786 Array used to obtain the complete symmetric function in n variables in terms of the elementary symmetric function. 7
1, -1, 1, 1, -2, 1, -1, 2, 1, -3, 1, 1, -2, -2, 3, 3, -4, 1, -1, 2, 2, 1, -3, -6, -1, 4, 6, -5, 1, 1, -2, -2, -2, 3, 6, 3, 3, -4, -12, -4, 5, 10, -6, 1, -1, 2, 2, 2, 1, -3, -6, -6, -3, -3, 4, 12, 6, 12, 1, -5, -20, -10, 6, 15, -7, 1, 1, -2, -2, -2, -2, 3, 6, 6, 3, 3, 6, 1, -4, -12, -12, -12, -12, -4, 5, 20, 10, 30, 5, -6, -30, -20, 7, 21, -8, 1, -1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

The unsigned numbers give A048996. They are not listed on p. 831-2 of Abramowitz and Stegun (reference given in A103921). One could call these numbers M_0 (like M_1, M_2, M_3 given in A036038, A036039, A036040, resp.).

The sequence of row lengths is A000041(n) (partition numbers).

The sign is (-1)^(n+m(n,k)) with m(n,k) the number of parts of the k-th partition of n taken in the mentioned order. For m(n,k) see A036043.

The row sum is 1 for n=1 and 0 otherwise. The unsigned row sum is 2^(n-1)=A000079(n-1), n>=1.

The complete symmetric polynomial is also h(n;a[1],...,a[n]) = Det A_n with the matrix elements of the n X n matrix A_n given by A_n(k,k+1)=1, A(k,m)=a[k-m+1],n>= k>=m>=1 and 0 else.

REFERENCES

P. A. MacMahon, Combinatory Analysis, 2 vols., Chelsea, NY, 1960, see p. 4.

V. Krishnamurthy, Combinatorics, Ellis Horwood, Chichester, 1986, p. 55, eqs. (48) and (50).

LINKS

Table of n, a(n) for n=1..97.

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].

P. A. MacMahon, Combinatory analysis.

Wolfdieter Lang, First 10 rows.

FORMULA

The complete symmetric row polynomials h(n;a[1], ..., a[n]):= sum k over partitions of n of a(n, k)* A[k}, with A[k}:=a[1]^e(k, 1)*a[2]^e(k, 2)*..*a[n]^e(k, n) if the k-th partition of n, in Abramowitz-Stegun order (see A105805 for this reference), is [1^e(k, 1), 2^e(k, 2), ..., n^e(k, n)], for k=1..p(n):= A000041(n) (partition numbers).

G.f.: A(x):=1/(1 + sum(((-1)^j)*a[j]*x^j, j=1..infinity).

a(n, k) is the coefficient of x^n and a[1]^e(k, 1)*a[2]^e(k, 2)*...*a[n]^e(k, n) in A(x) if the k-th partition of n, counted in Abramowitz-Stegun order, is [1^e(k, 1), 2^e(k, 2), ..., n^e(k, n)] with e(k, j)>=0 and if e(k, j)=0 then j^0 is not recorded.

a(n, k)= ((-1)^(n+m(n, k)))*m(n, k)!/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. m(n, k) is the number of parts of the k-th partition of n. See A036043 for m(n, k).

EXAMPLE

[1]; [ -1,1]; [1,-2,1]; [ -1,2,1,-3,1]; [1,-2,-2,3,3,-4,1]; ...

h(4;a[1],...,a[4])= -1*a[4] + 2*a[1]*a[3] + 1* a[2]^2 - 3*a[1]^2*a[2] + a[1]^4.

CROSSREFS

Sequence in context: A210961 A250007 A048996 * A072811 A296559 A233548

Adjacent sequences:  A111783 A111784 A111785 * A111787 A111788 A111789

KEYWORD

sign,tabf

AUTHOR

Wolfdieter Lang, Aug 23 2005

STATUS

approved

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Last modified March 22 04:32 EDT 2019. Contains 321406 sequences. (Running on oeis4.)