A196843 Table of the elementary symmetric functions a_k(1,2,3,5,6...n+1) (missing 4).

1, 1, 1, 1, 3, 2, 1, 6, 11, 6, 1, 11, 41, 61, 30, 1, 17, 107, 307, 396, 180, 1, 24, 226, 1056, 2545, 2952, 1260, 1, 32, 418, 2864, 10993, 23312, 24876, 10080, 1, 41, 706, 6626, 36769, 122249, 234684, 233964, 90720, 1, 51, 1116, 13686, 103029, 489939, 1457174

Offset

0,5

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 and x[j]=j+1 for j=4,...,n. This is the triangle S_4(n,k), n>=0, k=0..n. The first four rows coincide with those of triangle A094638.

Links

Table of n, a(n) for n=0..51.

Formula

a(n,k) = a_k(1,2,..,n) if 0<=n<4, and a_k(1,2,3,5,...,n+1) if n>=4, 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<4, and

a(n,k)= sum((-4)^m*|s(n+2,n+2-k+m)|,m=0..k) if n>=4

with the Stirling numbers of the first kind s(n,m)=

A048994(n,m).

Example

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  11   41   61    30
5:   1  17  107  307   396    180
6:   1  24  226 1056  2545   2952   1260
7:   1  32  418 2864 10993  23312  24876  10080
...
a(3,0) = a_0(1,2,3):= 1, a(3,1) = a_1(1,2,3)= 6.
a(4,2) = a_2(1,2,3,5) = 1*2+1*3+1*5+2*3+2*5+3*5 = 41.
a(4,2) = 1*|s(6,4)| - 4*|s(6,5)| + 16*|s(6,6)| =
  1*85 -4*15+16*1 = 41.

Crossrefs

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

Sequence in context: A008276 A094638 A196844 * A367023 A143778 A200536

Adjacent sequences: A196840 A196841 A196842 * A196844 A196845 A196846

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Oct 25 2011

Status

approved