[seqfan] conjectured EGF and extension of A086365 and others

[Joerg Arndt]

default(seriesprecision,66) \\ this many terms

t(i)=Vec(serlaplace(exp(-x+sum(j=1,i,((exp(j*x)-1)/j)))))
A000296:
 t(1)
[1, 0, 1, 1, 4, 11, 41, 162, 715, ...

A086365:
t(2)
[1, 1, 4, 15, 75, 428, 2781, 20093, ...
(could add more terms to OEIS entry)

t(3)
[1, 2, 10, 58, 416, 3450, 32356, 336838 ... not in OEIS (add?)


The function(s) t(i) are a slight variation of EGFs b(i) for
(max-increment) restricted growth strings (cf. section 15.3.4
of the fxtbook, the b(i) are one particular generalization of
the Bell numbers):

b(i)=Vec(serlaplace(exp(+x+sum(j=1,i,((exp(j*x)-1)/j)))))  \\ note plus
\\                       ^--= there

A000110 (Bell numbers):
 b(1)
[1, 2, 5, 15, 52, 203, 877, 4140, ...

A080337:
b(2)
[1, 3, 12, 59, 339, 2210, 16033, ...

none of the following in OEIS (add one or two?)
b(3)
[1, 4, 22, 150, 1200, 10922, ...

b(4)
[1, 5, 35, 305, 3125, 36479, 475295, ... 

b(5)
[1, 6, 51, 541, 6756, 96205, ...