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%I #64 Sep 07 2022 04:06:44
%S 1,6,72,1440,43200,1814400,101606400,7315660800,658409472000,
%T 72425041920000,9560105533440000,1491376463216640000,
%U 271430516305428480000,57000408424139980800000,13680098021793595392000000,3720986661927857946624000000
%N a(n) = n!*(n+1)!/2.
%C Column 2 in triangle A009963.
%C a(n) = A078740(n, 2), first column of (3, 2)-Stirling2 array.
%C Also the number of undirected Hamiltonian paths in the complete bipartite graph K_{n,n+1}. - _Eric W. Weisstein_, Sep 03 2017
%C Also, the number of undirected Hamiltonian cycles in the complete bipartite graph K_{n+1,n+1}. - _Pontus von Brömssen_, Sep 06 2022
%H Vincenzo Librandi, <a href="/A010796/b010796.txt">Table of n, a(n) for n = 1..200</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HamiltonianPath.html">Hamiltonian Path</a>.
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.
%F a(n) = 2^(n-1) * A006472(n+1).
%F a(n) = A010790(n)/2.
%F E.g.f.: (hypergeom([1, 2], [], x)-1)/2.
%F a(n) = Product_{k=1..n-1} (k^2+3*k+2). - _Gerry Martens_, May 09 2016
%F E.g.f.: x*hypergeom([1, 3], [], x). - _Robert Israel_, May 09 2016
%F From _Amiram Eldar_, Jun 25 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 2*(BesselI(1, 2) - 1).
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(1 - BesselJ(1, 2)). (End)
%t Table[n! (n + 1)! / 2, {n, 1, 20}] (* _Vincenzo Librandi_, Jun 11 2013 *)
%t Times@@@Partition[Range[20]!,2,1]/2 (* _Harvey P. Dale_, Jul 04 2017 *)
%o (Magma) [Factorial(n)* Factorial(n+1) / 2: n in [1..20]]; // _Vincenzo Librandi_, Jun 11 2013
%o (PARI) for(n=1,30, print1(n!*(n+1)!/2, ", ")) \\ _G. C. Greubel_, Feb 07 2018
%Y Cf. A006472, A009963, A078740, A010790.
%Y Main diagonal of A291909.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
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