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

Offset: 0

Author

Wolfdieter Lang, Oct 24 2011

Keywords

• nonn
• easy
• tabl

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

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.

Examples

			n\k  0   1   2    3     4      5      6      7  ...
0:   1
1:   1   1
2:   1   4   3
3:   1   8  19   12
4:   1  13  59  107    60
5:   1  19 137  461   702    360
6:   1  26 270 1420  3929   5274   2520
7:   1  34 478 3580 15289  36706  44712  20160
...
a(3,2) = 1*3+1*4+3*4 = 19.
a(3,2) = |s(5,3)| - 2*|s(5,4)| + 4*|s(5,5)| = 35-2*10+4*1 = 19.
		

References

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

Crossrefs

Cf. A094638, A145324,|A123319|.

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

= 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).

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