cp's OEIS Frontend

A196846 Table of elementary symmetric functions a_k(1,2,5,6,...,n+2) (no 3,4).

%I A196846 #7 Mar 30 2012 18:49:34
%S A196846 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 A196846 1871,5380,7172,3360,1,38,592,4850,22219,55592,67908,30240,1,48,972,
%U A196846 10770,70719,277782,623828,709320,302400,1,59,1500,21462,189189,1055691,3679430,7571428,8104920,3326400
%N A196846 Table of elementary symmetric functions a_k(1,2,5,6,...,n+2) (no 3,4).
%C A196846 For the symmetric functions a_k see a comment in A196841.
%C A196846 The definition of the family of number triangles
%C A196846 S_{i,j}(n,k),n>=k>=0, 1<=i<j<=n+2, has been given in
%C A196846 A196845. The present triangle is S_{3,4}(n,k) (no 3 and 4
%C A196846 admitted). The first three lines coincide with those of
%C A196846 triangle A094638(n+1,k+1) which tabulates a_k(1,2,...,n).
%F A196846 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 A196846 a(n,k) = |s(n+1,n+1-k)| for 0<=n<3,
%F A196846 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 A196846 n\k   0    1    2     3      4      5      6       7 ...
%e A196846 0:    1
%e A196846 1:    1    1
%e A196846 2:    1    3    2
%e A196846 3:    1    8   17    10
%e A196846 4:    1   14   65   112     60
%e A196846 5:    1   21  163   567    844    420
%e A196846 6:    1   29  331  1871   5380   7172   3360
%e A196846 7:    1   38  592  4850  22219  55592  67908   30240
%e A196846 ...
%e A196846 a(2,2)=a_2(1,2)=A094638(3,3)=1*2=2.
%e A196846 a(2,2) = |s(3,1)| = 2.
%e A196846 a(4,2) = a_2(1,2,5,6) = 1*2+1*5+1*6+2*5+2*6+5*6 = 65.
%e A196846 a(4,2) = 1*(|s(7,5)| - (3*S_3(5,1) + 4*S_4(5,1))) +
%e A196846 3*4*(|s(7,7)| -(3*0 + 4*0)) = 1*(175 -(3*18 + 4*17))
%e A196846 + 12*1 = 65.
%Y A196846 Cf. A094638 (a_k triangle), A196845 (no 1,2 triangle), A196842 (no 3), A196843 (no 4).
%K A196846 nonn,easy,tabl
%O A196846 0,5
%A A196846 _Wolfdieter Lang_, Oct 27 2011