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User:Peter Luschny/EulerianNumber

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ECT Encyclopedia Combinatorial Triangles

 

 

Eulerian Number

OEIS-Reference: A173018

Number of permutations of {1,2,..,n} with k ascents.


T(n,k) = \sum_{j=0}^k(-1)^j\binom{n+1}{j}(k-j+1)^n

 
Triangle-Array
1  
1 0  
1 1 0  
1 4 1 0  
1 11 11 1 0  
1 26 66 26 1 0  
1 57 302 302 57 1 0
 
sum als gcd lcm
1 1 0 1
1 1 0 1
2 0 0 1
6 2 4 4
24 0 11 11
120 16 2 858
720 0 1 17214
Rectangle-Array
1 1 1 1 1 1 1
0 1 4 11 26 57 120
0 1 11 66 302 1191 4293
0 1 26 302 2416 15619 88234
0 1 57 1191 15619 156190 1310354
0 1 120 4293 88234 1310354 15724248
0 1 247 14608 455192 9738114 162512286
Fingerprint
SubSeqType 0 1 2 3
Row A000012A000295A000460A000505
Column A000007A000012A000295A000460
RowDiag A180056A000000A000000A000000
ColDiag A180056A025585A000000A000000
Characteristic SUM ALTSUM LCM GCD
  A000142A009006A180057A000000

Maple T := proc(n,k) local j; add((-1)^j*(1+k-j)^n*binomial(n+1,j),j=0..k) end:

TeX T(n,k) = \sum_{j=0}^k(-1)^j\binom{n+1}{j}(k-j+1)^n

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