A196841 - OEIS

A196841

Table of the elementary symmetric functions a_k(1,3,4,...,n+1).

1, 1, 1, 1, 4, 3, 1, 8, 19, 12, 1, 13, 59, 107, 60, 1, 19, 137, 461, 702, 360, 1, 26, 270, 1420, 3929, 5274, 2520, 1, 34, 478, 3580, 15289, 36706, 44712, 20160, 1, 43, 784, 7882, 47509, 174307, 375066, 422568, 181440, 1, 53, 1214, 15722, 126329, 649397

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OFFSET

0,5

COMMENTS

The elementary symmetric functions are defined by product(1-x[j]*x,j=1..n)=: sum((-1)^k*a_k(x[1],x[2],...,x[n])*x^k ,k=0..n), n>=1. Here x[1]=1 and x[j]=j+1 for j=2,..,n.

This triangle is the row reversed version of |A123319|.

In general, the triangle S_j(n,k), lists for n>=j the elementary symmetric functions

a_k(1,2,...,j-1,j+1,...,n+1), k=0..n. For 0<=n<j one takes a_k(1,2,..,n), k=0..n, with a_0():=1.

For j=0 one takes a_0(n,k) = a_k(1,2,...,n) which is A094638(n+1,k+1). a_1(n,k)=a_k(2,3,....,n+1)= A145324(n+1,k+1). The present triangle a(n,k) equals S_2(n,k).

The first j rows of the triangle S_j(n,k) coincide with the ones of triangle A094638.

The following rows (n>=j) of S_j(n,k) are given by

sum((-j)^m*|s(n+2,n+2-k+m)|,m=0..k), with the Stirling numbers of the first kind s(n,m) = A048994(n,m). The proof is done by iterating the obvious recurrence S_j(l,m) = a_m(1,2,...,l+1) - j*S_j(l,m-1), using a_k(1,2,...,n) = |s(n+1,n+1-m)|, For a proof of the last equation see, e.g., the Stanley reference, p. 19, Second Proof.

REFERENCES

R. P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, 1997.

LINKS

Table of n, a(n) for n=0..50.

FORMULA

a(n,k) = a_k(1,2,..,n) if 0<=n<2, and a_k(1,3,4,...,n+1) if n>=2, for k=0..n, with the elementary symmetric functions a_k defined above in a comment.

a(n,k) = 0 if n<k, = |s(n+1,n+1-k)| if 0<=n<2, and

= sum((-2)^m*|s(n+2,n+2-k+m)|,m=0..k) if n>=2, with the Stirling numbers of the first kind s(n,m) = A048994(n,m).

EXAMPLE

n\k 0 1 2 3 4 5 6 7 ...

0: 1

1: 1 1