[seqfan] conjectured EGF and extension of A086365 and others
Joerg Arndt
arndt at jjj.de
Thu Apr 15 17:34:39 CEST 2010
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, ...
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