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A196844 Table of the elementary symmetric functions a_k(1,2,3,4,6,...,n+1) (5 missing). 0
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 (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, 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.
LINKS
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).
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 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
Sequence in context: A181853 A008276 A094638 * A196843 A367023 A143778
KEYWORD
nonn,easy,tabl
AUTHOR
Wolfdieter Lang, Oct 25 2011
STATUS
approved

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Last modified March 29 04:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)