OFFSET
0,3
COMMENTS
a(n) = A184178(n,0).
FORMULA
a(n) = Sum_{j=0..n} d(n-j)*Sum_{m=0..floor(j/2)} binomial(j-m-1, m-1)*binomial(n+1-j, m), where d(i) = A000166(i) are the derangement numbers.
EXAMPLE
a(3)=3 because we have 123, 231, and 312. The permutations (1)32, 21(3), and 3(2)1 do have isolated fixed points (shown between parentheses).
MAPLE
d[0] := 1: d[1] := 1: for n to 50 do d[n] := n*d[n-1]+(-1)^n end do: a := proc (n) options operator, arrow: add(d[n-j]*add(binomial(j-m-1, m-1)*binomial(n+1-j, m), m = 0 .. floor((1/2)*j)), j = 0 .. n) end proc: seq(a(n), n = 0 .. 22);
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Feb 13 2011
STATUS
approved