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

Original entry on oeis.org

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

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Author

Wolfdieter Lang, Oct 25 2011

Keywords

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.

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  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.

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
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).

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