%I #7 Mar 30 2012 18:49:34
%S 1,1,1,1,3,2,1,8,17,10,1,14,65,112,60,1,21,163,567,844,420,1,29,331,
%T 1871,5380,7172,3360,1,38,592,4850,22219,55592,67908,30240,1,48,972,
%U 10770,70719,277782,623828,709320,302400,1,59,1500,21462,189189,1055691,3679430,7571428,8104920,3326400
%N Table of elementary symmetric functions a_k(1,2,5,6,...,n+2) (no 3,4).
%C For the symmetric functions a_k see a comment in A196841.
%C The definition of the family of number triangles
%C S_{i,j}(n,k),n>=k>=0, 1<=i<j<=n+2, has been given in
%C A196845. The present triangle is S_{3,4}(n,k) (no 3 and 4
%C admitted). The first three lines coincide with those of
%C triangle A094638(n+1,k+1) which tabulates a_k(1,2,...,n).
%F a(n,k) = 0 if n<k, a(0,0) = 1, a(1,k) = a_k(1) for k=0,1, a(2,k) = a_k(1,2) for k=0,1,2, and a(n,k) = a_k(1,2,5,6,...,n+2), n>=3; k=0..n, with the elementary symmetric functions a_k (see the comment above).
%F a(n,k) = |s(n+1,n+1-k)| for 0<=n<3,
%F a(n,k) = sum(((3*4)^m)*(|s(n+3,n+3-k+2*m)| - (3*S_3(n+1,k-1-2*m) + 4*S_4(n+1,k-1-2*m))),m = 0..floor(k/2)), with the Stirling numbers of the first kind s(n,m) = A048994(n,m), and the number triangles S_3(n,k)= A196842(n,k) and S_4(n,k)= A196843(n,k) (for negative k one puts the entries of these triangles to 0).
%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 8 17 10
%e 4: 1 14 65 112 60
%e 5: 1 21 163 567 844 420
%e 6: 1 29 331 1871 5380 7172 3360
%e 7: 1 38 592 4850 22219 55592 67908 30240
%e ...
%e a(2,2)=a_2(1,2)=A094638(3,3)=1*2=2.
%e a(2,2) = |s(3,1)| = 2.
%e a(4,2) = a_2(1,2,5,6) = 1*2+1*5+1*6+2*5+2*6+5*6 = 65.
%e a(4,2) = 1*(|s(7,5)| - (3*S_3(5,1) + 4*S_4(5,1))) +
%e 3*4*(|s(7,7)| -(3*0 + 4*0)) = 1*(175 -(3*18 + 4*17))
%e + 12*1 = 65.
%Y Cf. A094638 (a_k triangle), A196845 (no 1,2 triangle), A196842 (no 3), A196843 (no 4).
%K nonn,easy,tabl
%O 0,5
%A _Wolfdieter Lang_, Oct 27 2011
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