1,3

a(n) = number of endofunctions f of [n] which interchange a pair a<->b and for all x in [n] some iterate f^k(x) = a. E.g. a(3) = 6: 1<->2<-3; 3->1<->2; 2<->3<-1; 1->2<->3; 1<->3<-2; 2->1<->3. - Len Smiley (smiley(AT)math.uaa.alaska.edu), Nov 27 2001

If offset is 0: right side of the binomial sum n-> sum( i^(i-1) * (n-i+1)^(n-i)*binomial(n, i), i=1..n) - Yong Kong (ykong(AT)curagen.com), Dec 28 2000

A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (4.2.2.36)

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Prop. 5.3.2.

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Eric Weisstein's World of Mathematics, Graph Edge

E.g.f.: 1/2!*LambertW(-x)^2. - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 07 2001

E.g.f. if offset 0: W(-x)^2/((1+W(-x))*x), W(x) Lambert's function (principal branch).

The sequence 1, 1, 6, 48 ... satisfies a(n)=(n(n+1)^n+0^n)/(n+1); it is the main diagonal of A085388. - Paul Barry (pbarry(AT)wit.ie), Jun 30 2003

Cf. A000169, A000312, A053506-A053509.

Sequence in context: A105627 A051578 A052639 this_sequence A055861 A052567 A002170

Adjacent sequences: A053503 A053504 A053505 this_sequence A053507 A053508 A053509

nonn

N. J. A. Sloane (njas(AT)research.att.com), Jan 15 2000