A196845 Table of elementary symmetric function a_k(3,4,...,n+2) (no 1 and 2).

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

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).

Links

Table of n, a(n) for n=0..54.

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).

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).

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

Cf. A196841, A048994, A145324, A001710 (diagonal), A001711 (1st subdiagonal), A001712 (2nd subdiagonal), A055998 (k=1), A024183 (k=2), A024184 (k=3), A024185 (k=4).

Sequence in context: A307901 A232965 A249401 * A263446 A297192 A218592

Adjacent sequences: A196842 A196843 A196844 * A196846 A196847 A196848

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Oct 26 2011

Status

approved