OFFSET
1,2
COMMENTS
Column 2 in triangle A009963.
a(n) = A078740(n, 2), first column of (3, 2)-Stirling2 array.
Also the number of undirected Hamiltonian paths in the complete bipartite graph K_{n,n+1}. - Eric W. Weisstein, Sep 03 2017
Also, the number of undirected Hamiltonian cycles in the complete bipartite graph K_{n+1,n+1}. - Pontus von Brömssen, Sep 06 2022
For n >= 3, a(n) is also the denominator of the likelihood of the (n+1)-star graph S_(n+1), in lowest terms. - Eric W. Weisstein, Jul 31 2026
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..200
El-Mehdi Mehiri, Bijections Between Smirnov Words and Hamiltonian Cycles in Complete Multipartite Graphs, arXiv:2510.26597 [math.CO], 2025. See p. 10. Table 1.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
Eric Weisstein's World of Mathematics, Graph Likelihood.
Eric Weisstein's World of Mathematics, Hamiltonian Cycle.
Eric Weisstein's World of Mathematics, Hamiltonian Path.
Eric Weisstein's World of Mathematics, Star Graph.
FORMULA
a(n) = 2^(n-1) * A006472(n+1).
a(n) = A010790(n)/2.
E.g.f.: (hypergeom([1, 2], [], x)-1)/2.
a(n) = Product_{k=1..n-1} (k^2+3*k+2). - Gerry Martens, May 09 2016
E.g.f.: x*hypergeom([1, 3], [], x). - Robert Israel, May 09 2016
From Amiram Eldar, Jun 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 2*(BesselI(1, 2) - 1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(1 - BesselJ(1, 2)). (End)
MATHEMATICA
Table[n! (n + 1)! / 2, {n, 1, 20}] (* Vincenzo Librandi, Jun 11 2013 *)
Times@@@Partition[Range[20]!, 2, 1]/2 (* Harvey P. Dale, Jul 04 2017 *)
PROG
(Magma) [Factorial(n)* Factorial(n+1) / 2: n in [1..20]]; // Vincenzo Librandi, Jun 11 2013
(PARI) for(n=1, 30, print1(n!*(n+1)!/2, ", ")) \\ G. C. Greubel, Feb 07 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved