[seqfan] A055814: permutations with all cycles of length 2 or 3

Joerg Arndt arndt at jjj.de
Fri Oct 2 05:21:01 CEST 2009

In  http://www.research.att.com/~njas/sequences/A055814
I suggest to change the title to something like
given in the subject line.

The current title
 "Taylor coefficients at x=0 of exp(x^3/3+x^2/2)."
may be incorrect (pari/gp):
? exp(x^3/3+x^2/2)
1 + 1/2*x^2 + 1/3*x^3 + 1/8*x^4 + 1/6*x^5 + ...
\\ The function is really the EGF:
? serlaplace(exp(x^3/3+x^1/1))
1 + x + x^2 + 3*x^3 + 9*x^4 + 21*x^5 + ...

Also (please check) we should have
 a(n)=(n-1)*a(n-2) + (n-1)*(n-2)*a(n-3)
with inital values a(-n)=0, a(0)=1

Additionally we may want to note that
the EGF for perms with cycles \in {l1,l2,...,lk}
is exp( sum(j=1..k, x^k/k) )
and the recursion is (same inital values)
a(n)= sum(j=1..k, F(n-1,j-1)*a(n-j) )
where F(n-1,j):=(n-1)*(n-2)*...*(n-j+1)


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