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

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 * A143778 A200536 A164645

Adjacent sequences:  A196840 A196841 A196842 * A196844 A196845 A196846

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Oct 25 2011

STATUS

approved

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Last modified February 24 17:22 EST 2018. Contains 299624 sequences. (Running on oeis4.)