A196844 Table of the elementary symmetric functions a_k(1,2,3,4,6,...,n+1) (5 missing).

1, 1, 1, 1, 3, 2, 1, 6, 11, 6, 1, 10, 35, 50, 24, 1, 16, 95, 260, 324, 144, 1, 23, 207, 925, 2144, 2412, 1008, 1, 31, 391, 2581, 9544, 19564, 20304, 8064, 1, 40, 670, 6100, 32773, 105460, 196380, 190800, 72576, 1, 50, 1070, 12800, 93773, 433190, 1250980

Offset: 0

Author

Wolfdieter Lang, Oct 25 2011

Keywords

• nonn
• easy
• tabl

Comments

For the symmetric functions a_k and the definition of the triangles S_j(n,k) see a comment in A196841. Here x(j) = j for j = 1, 2, 3, 4 and x(j) = j + 1 for j = 5, ..., n. This is the triangle S_5(n,k), n >= 0, k = 0..n. The first five rows coincide with those of triangle A094638.

Examples

			n\k 0   1    2     3     4      5      6     7 ...
0:  1
1:  1   1
2:  1   3    2
3:  1   6   11     6
4:  1  10   35    50    24
5:  1  16   95   260   324    144
6:  1  23  207   925  2144   2412   1008
7:  1  31  391  2581  9544  19564  20304  8064
...
a(4,0) = a_0(1, 2, 3, 4) := 1, a(4,1) = a_1(1, 2, 3, 4) = 10.
a(5,2) = a_2(1, 2, 3, 4, 6) = 1*2 + 1*3 + 1*4 + 1*6 + 2*3 + 2*4 + 2*6 + 3*4 + 3*6 + 4*6 = 95.
a(5,2) = 1*|s(7,5)| - 5*|s(7,6)| + 25*|s(7,7)| = 1*175 - 5*21 + 25*1 = 95.
		

Crossrefs

Cf. A094638, A145324,|A123319|, A196841, A196842, A196843.

Formula

a(n,k) = a_k(1, 2, ..., n) if 0 <= n < 5, and a_k(1, 2, 3, 4, 6, 7, ..., n+1) if n >= 5, with the elementary symmetric functions a_k defined in a comment to A196841.

a(n,k) = 0 if n < k, a(n,k) = |s(n+1, n+1-k)| if 0 <= n < 5, and

a(n,k) = sum((-5)^m*|s(n+2, n+2-k+m)|, m = 0..k) if n >= 5, with the Stirling numbers of the first kind s(n,m)=A048994(n,m).