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User:Peter Luschny/EulerianNumber

ECT Encyclopedia Combinatorial Triangles

Eulerian Number

OEIS-Reference: A173018

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

${\displaystyle T(n,k)=\sum _{j=0}^{k}(-1)^{j}{\binom {n+1}{j}}(k-j+1)^{n}}$

 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 A000012 A000295 A000460 A000505 Column A000007 A000012 A000295 A000460 RowDiag A180056 A000000 A000000 A000000 ColDiag A180056 A025585 A000000 A000000 Characteristic SUM ALTSUM LCM GCD A000142 A009006 A180057 A000000

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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