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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) 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 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=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.)