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A196841 Table of the elementary symmetric functions a_k(1,3,4,...,n+1). 5
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 (list; table; graph; refs; listen; history; text; internal format)
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

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.

CROSSREFS

Cf. A094638, A145324,|A123319|.

Sequence in context: A165914 A139621 A305621 * A165732 A197698 A193011

Adjacent sequences:  A196838 A196839 A196840 * A196842 A196843 A196844

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Oct 24 2011

STATUS

approved

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Last modified April 16 23:40 EDT 2021. Contains 343051 sequences. (Running on oeis4.)