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A196846
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Table of elementary symmetric functions a_k(1,2,5,6,...,n+2) (no 3,4).
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2
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1, 1, 1, 1, 3, 2, 1, 8, 17, 10, 1, 14, 65, 112, 60, 1, 21, 163, 567, 844, 420, 1, 29, 331, 1871, 5380, 7172, 3360, 1, 38, 592, 4850, 22219, 55592, 67908, 30240, 1, 48, 972, 10770, 70719, 277782, 623828, 709320, 302400, 1, 59, 1500, 21462, 189189, 1055691, 3679430, 7571428, 8104920, 3326400
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OFFSET
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0,5
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COMMENTS
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For the symmetric functions a_k see a comment in A196841.
The definition of the family of number triangles
S_{i,j}(n,k),n>=k>=0, 1<=i<j<=n+2, has been given in
A196845. The present triangle is S_{3,4}(n,k) (no 3 and 4
admitted). The first three lines coincide with those of
triangle A094638(n+1,k+1) which tabulates a_k(1,2,...,n).
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LINKS
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FORMULA
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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).
a(n,k) = |s(n+1,n+1-k)| for 0<=n<3,
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).
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EXAMPLE
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n\k 0 1 2 3 4 5 6 7 ...
0: 1
1: 1 1
2: 1 3 2
3: 1 8 17 10
4: 1 14 65 112 60
5: 1 21 163 567 844 420
6: 1 29 331 1871 5380 7172 3360
7: 1 38 592 4850 22219 55592 67908 30240
...
a(2,2)=a_2(1,2)=A094638(3,3)=1*2=2.
a(2,2) = |s(3,1)| = 2.
a(4,2) = a_2(1,2,5,6) = 1*2+1*5+1*6+2*5+2*6+5*6 = 65.
a(4,2) = 1*(|s(7,5)| - (3*S_3(5,1) + 4*S_4(5,1))) +
3*4*(|s(7,7)| -(3*0 + 4*0)) = 1*(175 -(3*18 + 4*17))
+ 12*1 = 65.
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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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