A212559 Number of functions f:{1,2,...,n}->{1,2,...,n} such that every non-recurrent element has at most one preimage.

1, 1, 4, 27, 244, 2745, 36966, 580111, 10399096, 209672721, 4696872490, 115732052271, 3110867569140, 90587751885241, 2840805169411678, 95450112571474095, 3420897993621996016, 130266500391456691233, 5252293203395848789842, 223535386151669737094095, 10014286301754519472897900

Offset: 0

Author

Geoffrey Critzer, May 21 2012

Keywords

• nonn

Comments

An element x of {1,2,...,n} is a recurrent element if there exists a positive integer k such that (f^k)(x) = x where f^k is the k-th iteration of functional composition.

The functional digraphs are composed of cycles of rooted trees with every non-root vertex of degree 1 or 2. Cf. A006152.

Programs

Mathematica

nn=20;a=x Exp[x/(1-x)];Range[0,nn]! CoefficientList[Series[1/(1-a),{x,0,nn}],x]

Formula

E.g.f.: 1/(1-A(x)) where A(x) is the e.g.f. for A006152.