1,5

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.

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).

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.

W. Lang, First 10 rows.

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) (partititon 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).

[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.

Sequence in context: A153905 A165357 A048996 this_sequence A072811 A080027 A050305

Adjacent sequences: A111783 A111784 A111785 this_sequence A111787 A111788 A111789

sign,tabf

Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Aug 23 2005