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Table of the elementary symmetric functions a_k(1,2,3,5,6...n+1) (missing 4).
2

%I #12 Apr 02 2014 19:18:25

%S 1,1,1,1,3,2,1,6,11,6,1,11,41,61,30,1,17,107,307,396,180,1,24,226,

%T 1056,2545,2952,1260,1,32,418,2864,10993,23312,24876,10080,1,41,706,

%U 6626,36769,122249,234684,233964,90720,1,51,1116,13686,103029,489939,1457174

%N Table of the elementary symmetric functions a_k(1,2,3,5,6...n+1) (missing 4).

%C For the symmetric functions a_k and the definition of the triangles S_j(n,k) see a comment in A196841. Here x[j]=j for j=1,2,3 and x[j]=j+1 for j=4,...,n. This is the triangle S_4(n,k), n>=0, k=0..n. The first four rows coincide with those of triangle A094638.

%F a(n,k) = a_k(1,2,..,n) if 0<=n<4, and a_k(1,2,3,5,...,n+1) if n>=4, with the elementary symmetric functions a_k defined in a comment to A196841.

%F a(n,k) = 0 if n<k, a(n,k)= |s(n+1,n+1-k)| if 0<=n<4, and

%F a(n,k)= sum((-4)^m*|s(n+2,n+2-k+m)|,m=0..k) if n>=4

%F with the Stirling numbers of the first kind s(n,m)=

%F A048994(n,m).

%e n\k 0 1 2 3 4 5 6 7 ...

%e 0: 1

%e 1: 1 1

%e 2: 1 3 2

%e 3: 1 6 11 6

%e 4: 1 11 41 61 30

%e 5: 1 17 107 307 396 180

%e 6: 1 24 226 1056 2545 2952 1260

%e 7: 1 32 418 2864 10993 23312 24876 10080

%e ...

%e a(3,0) = a_0(1,2,3):= 1, a(3,1) = a_1(1,2,3)= 6.

%e a(4,2) = a_2(1,2,3,5) = 1*2+1*3+1*5+2*3+2*5+3*5 = 41.

%e a(4,2) = 1*|s(6,4)| - 4*|s(6,5)| + 16*|s(6,6)| =

%e 1*85 -4*15+16*1 = 41.

%Y Cf. A094638, A145324,|A123319|, A196841, A196842.

%K nonn,easy,tabl

%O 0,5

%A _Wolfdieter Lang_, Oct 25 2011