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A196843 Table of the elementary symmetric functions a_k(1,2,3,5,6...n+1) (missing 4).

%I A196843 #12 Apr 02 2014 19:18:25
%S A196843 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 A196843 1056,2545,2952,1260,1,32,418,2864,10993,23312,24876,10080,1,41,706,
%U A196843 6626,36769,122249,234684,233964,90720,1,51,1116,13686,103029,489939,1457174
%N A196843 Table of the elementary symmetric functions a_k(1,2,3,5,6...n+1) (missing 4).
%C A196843 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 A196843 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 A196843 a(n,k) = 0 if n<k, a(n,k)= |s(n+1,n+1-k)| if 0<=n<4, and
%F A196843   a(n,k)= sum((-4)^m*|s(n+2,n+2-k+m)|,m=0..k) if n>=4
%F A196843   with the Stirling numbers of the first kind s(n,m)=
%F A196843   A048994(n,m).
%e A196843 n\k  0   1    2    3     4      5     6      7   ...
%e A196843 0:   1
%e A196843 1:   1   1
%e A196843 2:   1   3    2
%e A196843 3:   1   6   11    6
%e A196843 4:   1  11   41   61    30
%e A196843 5:   1  17  107  307   396    180
%e A196843 6:   1  24  226 1056  2545   2952   1260
%e A196843 7:   1  32  418 2864 10993  23312  24876  10080
%e A196843 ...
%e A196843 a(3,0) = a_0(1,2,3):= 1, a(3,1) = a_1(1,2,3)= 6.
%e A196843 a(4,2) = a_2(1,2,3,5) = 1*2+1*3+1*5+2*3+2*5+3*5 = 41.
%e A196843 a(4,2) = 1*|s(6,4)| - 4*|s(6,5)| + 16*|s(6,6)| =
%e A196843   1*85 -4*15+16*1 = 41.
%Y A196843 Cf. A094638, A145324,|A123319|, A196841, A196842.
%K A196843 nonn,easy,tabl
%O A196843 0,5
%A A196843 _Wolfdieter Lang_, Oct 25 2011