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A053506 a(n) = (n-1)*n^(n-2). 20
0, 1, 6, 48, 500, 6480, 100842, 1835008, 38263752, 900000000, 23579476910, 681091006464, 21505924728444, 737020860878848, 27246730957031250, 1080863910568919040, 45798768824157052688, 2064472028642102280192 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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, 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

REFERENCES

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.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..250

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Eric Weisstein's World of Mathematics, Graph Edge

FORMULA

E.g.f.: 1/2!*LambertW(-x)^2. - Vladeta Jovovic, 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, Jun 30 2003

a(n) = Sum_{i=1..n-1} binomial(n-1,i-1)*i^(i-2)*(n-i)^(n-i). - Dmitry Kruchinin, Oct 28 2013

If offset = 0 and a(0) = 1 then a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(-k,-n)*n^k (cf. A195242). - Peter Luschny, Apr 11 2016

MATHEMATICA

Table[(n-1)*n^(n-2), {n, 20}]

PROG

(PARI) for(n=1, 25, print1(binomial(n-1, 1)*n^(n-2), ", ")) \\ G. C. Greubel, Jan 18 2017

CROSSREFS

Cf. A000169, A000312, A053507, A053508, A053509, A195242.

Cf. A001865 which is the sum of A000169 + A053506 + A065513 + A065888 + ...

Sequence in context: A248744 A261900 * A055861 A052567 A002170 A153467

Adjacent sequences:  A053503 A053504 A053505 * A053507 A053508 A053509

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jan 15 2000

STATUS

approved

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Last modified February 23 12:50 EST 2018. Contains 299581 sequences. (Running on oeis4.)