OFFSET
0,3
COMMENTS
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).
FORMULA
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).
EXAMPLE
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.
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Oct 26 2011
STATUS
approved