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%I #8 Mar 30 2012 18:49:34
%S 1,1,1,1,4,3,1,8,19,12,1,13,59,107,60,1,19,137,461,702,360,1,26,270,
%T 1420,3929,5274,2520,1,34,478,3580,15289,36706,44712,20160,1,43,784,
%U 7882,47509,174307,375066,422568,181440,1,53,1214,15722,126329,649397
%N Table of the elementary symmetric functions a_k(1,3,4,...,n+1).
%C 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.
%C This triangle is the row reversed version of |A123319|.
%C In general, the triangle S_j(n,k), lists for n>=j the elementary symmetric functions
%C 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.
%C 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).
%C The first j rows of the triangle S_j(n,k) coincide with the ones of triangle A094638.
%C The following rows (n>=j) of S_j(n,k) are given by
%C 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.
%D R. P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, 1997.
%F 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.
%F a(n,k) = 0 if n<k, = |s(n+1,n+1-k)| if 0<=n<2, and
%F = 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).
%e n\k 0 1 2 3 4 5 6 7 ...
%e 0: 1
%e 1: 1 1
%e 2: 1 4 3
%e 3: 1 8 19 12
%e 4: 1 13 59 107 60
%e 5: 1 19 137 461 702 360
%e 6: 1 26 270 1420 3929 5274 2520
%e 7: 1 34 478 3580 15289 36706 44712 20160
%e ...
%e a(3,2) = 1*3+1*4+3*4 = 19.
%e a(3,2) = |s(5,3)| - 2*|s(5,4)| + 4*|s(5,5)| = 35-2*10+4*1 = 19.
%Y Cf. A094638, A145324,|A123319|.
%K nonn,easy,tabl
%O 0,5
%A _Wolfdieter Lang_, Oct 24 2011
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