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A196842
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Table of the elementary symmetric functions a_k(1,2,4,5,...,n+1).
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5
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1, 1, 1, 1, 3, 2, 1, 7, 14, 8, 1, 12, 49, 78, 40, 1, 18, 121, 372, 508, 240, 1, 25, 247, 1219, 3112, 3796, 1680, 1, 33, 447, 3195, 12864, 28692, 32048, 13440, 1, 42, 744, 7218, 41619, 144468, 290276, 301872, 120960, 1, 52, 1164, 14658, 113799, 560658, 1734956, 3204632, 3139680, 1209600
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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 and the definition of the triangles S_j(n,k) see a comment in A196841. Here x[1]=1, x[2]=2, and x[j]=j+1 for j=3,...,n. This is the triangle S_3(n,k), n>=0, k=0..n. The first three rows coincide with those of triangle A094638.
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LINKS
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FORMULA
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a(n,k) = a_k(1,2,..,n) if 0<=n<3, and a_k(1,2,4,5,...,n+1) if n>=3, with the elementary symmetric functions a_k defined in a comment to A196841.
a(n,k) = 0 if n<k, = |s(n+1,n+1-k)| if 0<=n<3, and= sum((-3)^m*|s(n+2,n+2-k+m)|,m=0..k) if n>=3, with the Stirling numbers of the first kind s(n,m)=A048994(n,m).
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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 7 14 8
4: 1 12 49 78 40
5: 1 18 121 372 508 240
6: 1 25 247 1219 3112 3796 1680
7: 1 33 447 3195 12864 28692 32048 13440
...
a(1,0) = a_0(1):= 1, a(1,1) = a_1(1)= 1.
a(3,2) = a_2(1,2,4) = 1*2 + 1*4 + 2*4 = 14.
a(3,2) = 1*|s(5,3)| - 3*|s(5,4)| + 9*|s(5,5)| = 1*35-3*10+9*1 = 14.
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MAPLE
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if n = 1 and k =1 then
1 ;
else
add( abs( combinat[stirling1](n+2, n+2-k+m))*(-3)^m, m=0..k) ;
end if;
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MATHEMATICA
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a[n_, k_] := If[n == 1 && k == 1, 1, Sum[(-3)^m Abs[StirlingS1[n + 2, n + 2 - k + m]], {m, 0, k}]];
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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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