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

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

Offset: 0

Author

Wolfdieter Lang, Oct 27 2011

Keywords

• nonn
• easy
• tabl

Comments

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

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).

Examples

			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.
		

Crossrefs

Cf. A094638 (a_k triangle), A196845 (no 1,2 triangle), A196842 (no 3), A196843 (no 4).

Formula

a(n,k) = 0 if 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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