cp's OEIS Frontend

A141834 Sum of the lengths of the cycles at a vertex of the complete graph K_n.

%I A141834 #2 Mar 31 2012 13:47:33
%S A141834 6,42,252,1620,11730,95886,876792,8877672,98640990,1193556210,
%T A141834 15624736116,220048367292,3317652307242,53319412081110,
%U A141834 909984632851440,16436597430879696,313262209859119542,6282647653285675962
%N A141834 Sum of the lengths of the cycles at a vertex of the complete graph K_n.
%C A141834 In graph theory, a complete graph is a simple graph in which every pair of distinct vertices is connected by an edge. The complete graph on n vertices is denoted by K_n. The number of simple cycles through a vertex of K_n equals A038154(n-1). See A000522 for the number of paths between a pair of distinct vertices of K_n.
%H A141834 Mehdi Hassani, <a href="http://arXiv.org/abs/math.CO/0606613">Counting and computing by e</a>
%F A141834 a(n) = floor(n!*e) - floor((n-1)!*e) - 2*n + 1 [Hassani]. a(n) = 2 - 2*n + (n-1)*(n-1)!*sum {k = 0..n-1} 1/k!. E.g.f.: sum {n = 0..inf} a(n+3)*x^n/n! = (6+12*x-21*x^2+8*x^3+3*x^4-2*x^5)*exp(x)/(1-x)^4 = 6 + 42*x + 252*x^2/2! + ... .
%e A141834 a(4) = 42. In the complete graph K_4 with vertex set {A,B,C,D} there are 12 simple cycles at the vertex A, namely the six 3-cycles ABCA, ABDA, ACBA, ACDA, ADBA and ADCA and the six 4-cycles ABCDA, ABDCA, ACBDA, ACDBA, ADBCA and ADCBA. The sum of the lengths of these cycles is 6*3 + 6*4 = 42.
%p A141834 a:= n -> 2 - 2*n + (n-1)*(n-1)!*add(1/k!,k = 0..n-1): seq(a(n), n = 3..22);
%Y A141834 Cf. A000522, A038154.
%K A141834 easy,nonn
%O A141834 3,1
%A A141834 _Peter Bala_, Jul 09 2008