A196843 - OEIS

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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

(list; table; graph; refs; listen; history; text; internal format)

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