Online Encyclopedia of Integer Sequences:

Bessel numbers: the number of nonoverlapping partitions of an n-set into equivalence classes.
Antidiagonal sums of nexus numbers (A047969).
a(0)=1; thereafter, a(m+1) = Sum_{k=0..m} k!*a(m-k).
Row sums of the triangle of triangular binomial coefficients given by A098568.
Number of permutations containing 3241 patterns only as part of 35241 patterns.
Number of permutations in S_n avoiding 21{bar 3}54 (i.e., every occurrence of 2154 is contained in an occurrence of a 21354).
Expansion of g.f.: A(x) = Product_{n>=0} 1/( 1 - x/(1-x)^n )^( 1/2^(n+1) ).
Number of permutations in S_n avoiding {bar 1}432 (i.e., every occurrence of 432 is contained in an occurrence of a 1432).
Number of permutations in S_n avoiding 352{bar 4}1 (i.e., every occurrence of 3521 is contained in an occurrence of a 35241).
Number of permutations in S_n avoiding 1{bar 4}235 (i.e., every occurrence of 1235 is contained in an occurrence of a 14235).
Number of permutations in S_n avoiding {bar 4}2153 (i.e., every occurrence of 2153 is contained in an occurrence of a 42153).
Number of permutations in S_n avoiding 5134{bar 2} (i.e., every occurrence of 5134 is contained in an occurrence of a 51342).
Number of permutations in S_n avoiding 25{bar 1}34 (i.e., every occurrence of 2534 is contained in an occurrence of a 25134).
Number of permutations in S_n avoiding 1{bar 5}324 (i.e., every occurrence of 1324 is contained in an occurrence of a 15324).
Number of permutations in S_n avoiding {bar 4}1253 (i.e., every occurrence of 1253 is contained in an occurrence of a 41253).
Number of permutations in S_n avoiding 1{bar 5}234 (i.e., every occurrence of 1234 is contained in an occurrence of a 15234).
Number of permutations in S_n avoiding {bar 1}3425 (i.e., every occurrence of 3425 is contained in an occurrence of a 13425).
Number of permutations in S_n avoiding {bar 1}3245 (i.e., every occurrence of 3245 is contained in an occurrence of a 13245).
Number of permutations in S_n avoiding {bar 5}1342 (i.e., every occurrence of 1342 is contained in an occurrence of a 51342).
Number of permutations in S_n avoiding {bar 5}2143 (i.e., every occurrence of 2143 is contained in an occurrence of a 52143).
Number of permutations in S_n avoiding 5234{bar 1} (i.e., every occurrence of 5234 is contained in an occurrence of a 52341).
Number of permutations in S_n avoiding {bar 5}1243 (i.e., every occurrence of 1243 is contained in an occurrence of a 51243).
Number of permutations in S_n avoiding {bar 5}1234 (i.e., every occurrence of 1234 is contained in an occurrence of a 51234).
Number of permutations in S_n avoiding 5{bar 1}{bar 2}43 (i.e., every occurrence of 543 is contained in an occurrence of a 51243).
Number of permutations in S_n avoiding 31{bar 5}{bar 4}2 (i.e., every occurrence of 312 is contained in an occurrence of a 31542).
Number of permutations in S_n avoiding {bar 2}413{bar 5} (i.e., every occurrence of 413 is contained in an occurrence of a 24135).
Number of permutations in S_n avoiding {bar 1}5{bar 2}43 (i.e., every occurrence of 543 is contained in an occurrence of 15243).
Number of permutations in S_n avoiding {bar 5}{bar 4}231 (i.e., every occurrence of 231 is contained in an occurrence of a 54231).
Number of permutations in S_n avoiding {bar 5}{bar 4}132 (i.e., every occurrence of 132 is contained in an occurrence of a 54132).
Number of permutations in S_n avoiding {bar 4}{bar 5}123 (i.e., every occurrence of 123 is contained in an occurrence of a 45123).
Number of permutations in S_n avoiding {bar 1}432{bar 5} (i.e., every occurrence of 432 is contained in an occurrence of a 14325).
Number of permutations in S_n avoiding 345{bar 2}{bar 1} (i.e., every occurrence of 345 is contained in an occurrence of a 34521).
Number of permutations that avoid the barred pattern bar{1}43bar{5}2.