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A196845
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Table of elementary symmetric function a_k(3,4,...,n+2) (no 1 and 2).
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3
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1, 1, 3, 1, 7, 12, 1, 12, 47, 60, 1, 18, 119, 342, 360, 1, 25, 245, 1175, 2754, 2520, 1, 33, 445, 3135, 12154, 24552, 20160, 1, 42, 742, 7140, 40369, 133938, 241128, 181440, 1, 52, 1162, 14560, 111769, 537628, 1580508, 2592720, 1814400, 1, 63, 1734, 27342, 271929, 1767087, 7494416, 19978308, 30334320, 19958400
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OFFSET
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0,3
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COMMENTS
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For the symmetric functions a_k see a comment in A196841.
In general the triangle S_{i,j}(n,k), n>=k>=0, 1<=i<j<=n+2 is defined for n<i as a_k(1,2,...,n), and for n>=i as a_k(1,2,...,i-1,i+1,...,j-1,j+1,...,n+2).
a_0():=1. The present triangle is S_{1,2}(n,k) (no 1 and 2 admitted).
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LINKS
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FORMULA
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a(n,k) = 0 if n<k, a(n,k) = a_k(3,4,...,n+2), n>=0, k=0,...,n, with the elementary symmetric function a_k (see the comment above).
a(n,k) = sum(2^k*( |s(n+3,n+3-k+2*p)| -(S_1(n+1,k-1-2*p) +2*S_2(n+1,k-1-2*p))), p=0..floor(k/2)), with the Stirling numbers of the first kind s(n,m) = A048994(n,m), and the number triangles S_1(n,k)= A145324(n+1,k+1) and S_2(n,k) = A196841(n,k).
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EXAMPLE
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n\k 0 1 2 3 4 5 6 7 ...
0: 1
1: 1 3
2: 1 7 12
3: 1 12 47 60
4: 1 18 119 342 360
5: 1 25 245 1175 2754 2520
6: 1 33 445 3135 12154 24552 20160
7: 1 42 742 7140 40369 133938 241128 181440
...
a(3,2) = a_2(3,4,5) = 3*4+3*5+4*5 = 47.
a(3,2) = 1*(|s(6,4)| - (1*14 + 2*13)) + 2*(|s(6,6)| -(1*0+2*0)) = 85 - 40 + 2(1-0) = 47.
a(4,3) = a_3(3,4,5,6) = 3*4*5+3*4*6+3*5*6+4*5*6 = 342.
a(4,3) = 1*(|s(7,4)| - (1*155 + 2*137)) + 2*(|s(7,6)| - (1*1 + 2*1)) = 735-429+2*(21-3) = 342.
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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