[seqfan] Bell numbers of an increasing unbounded positive sequence

[Vladimir Shevelev]

Dear SeqFans,

I submitted sequence A266946 "Smallest
numbers of distinct digital types of the positive
integers in base 10". If the base 10 in general to
change by base k, then let us denote by m_k(n) 
the number of different digital types of n-digit
numbers in base k. Then exactly the k first
values of m_k(n), n=1,...,k, coincide with
Bell numbers A110(1),...,A110(k). One can prove
this, using the comment in A110 by Bill Blewett.
Consider now an increasing unbounded posiive
sequence A={a(n)}. Then we can consider the 
smallest numbers of distinct digital types of A in
base k.  Denote by (m^A)_k(n) the number of
different digital types of n-digit numbers in A in
base k. If A contains a single-digit number, then
the union of {1,1} and  {(m^A)_k (k)} 
for k=2,3,4,... is naturally called the Bell numbers 
of sequence A. The first 1 formally corresponds
to n=0 in A110.  If A not contains any single-digit
number, then the union of {1,0} and  {(m^A)_k (k)} 
for k=2,3,4,... is Bell sequence for A.
 For example, let A be positive palindrome in
increasing order. Then, by this definition, the
corresponding Bell numbers are
1,1,1,2,2,5,5,15,15,... (cf. in the submitted A264406).

Best regards,
Vladimir