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A196841
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Table of the elementary symmetric functions a_k(1,3,4,...,n+1).
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5
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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
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0,5
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COMMENTS
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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.
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, 1997.
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LINKS
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FORMULA
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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).
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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