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%I #14 Apr 03 2014 03:11:51
%S 1,1,1,1,3,2,1,6,11,6,1,10,35,50,24,1,16,95,260,324,144,1,23,207,925,
%T 2144,2412,1008,1,31,391,2581,9544,19564,20304,8064,1,40,670,6100,
%U 32773,105460,196380,190800,72576,1,50,1070,12800,93773,433190,1250980
%N Table of the elementary symmetric functions a_k(1,2,3,4,6,...,n+1) (5 missing).
%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, 4 and x(j) = j + 1 for j = 5, ..., n. This is the triangle S_5(n,k), n >= 0, k = 0..n. The first five rows coincide with those of triangle A094638.
%F a(n,k) = a_k(1, 2, ..., n) if 0 <= n < 5, and a_k(1, 2, 3, 4, 6, 7, ..., n+1) if n >= 5, 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 < 5, and
%F a(n,k) = sum((-5)^m*|s(n+2, n+2-k+m)|, m = 0..k) if n >= 5, with the Stirling numbers of the first kind s(n,m)=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 10 35 50 24
%e 5: 1 16 95 260 324 144
%e 6: 1 23 207 925 2144 2412 1008
%e 7: 1 31 391 2581 9544 19564 20304 8064
%e ...
%e a(4,0) = a_0(1, 2, 3, 4) := 1, a(4,1) = a_1(1, 2, 3, 4) = 10.
%e a(5,2) = a_2(1, 2, 3, 4, 6) = 1*2 + 1*3 + 1*4 + 1*6 + 2*3 + 2*4 + 2*6 + 3*4 + 3*6 + 4*6 = 95.
%e a(5,2) = 1*|s(7,5)| - 5*|s(7,6)| + 25*|s(7,7)| = 1*175 - 5*21 + 25*1 = 95.
%Y Cf. A094638, A145324,|A123319|, A196841, A196842, A196843.
%K nonn,easy,tabl
%O 0,5
%A _Wolfdieter Lang_, Oct 25 2011
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