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%I #43 Jan 01 2019 06:36:39
%S 2,24,720,13824,383000,14804640,764340024,50913153536,4256161751448,
%T 436618291524000,53955264479804600,7908071556041000064,
%U 1356709951589099693976,269380212536429979520928,61297096735652845698099000,15847986814197933588682229760
%N Number of (directed) Hamiltonian paths in the graph join C_n + C_n of two cycles.
%C This graph is isomorphic to the circulant graph on 2n vertices where u,v are adjacent iff |u-v|=2 or |u-v| is odd. This sequence should not be confused with A268838, the number of Hamiltonian paths in C_n X C_n. The sequence can be computed through an analysis of the partitions of n. (See attached C# code for details). - _Andrew Howroyd_, Feb 14 2016
%H Andrew Howroyd and Vaclav Kotesovec, <a href="/A137891/b137891.txt">Table of n, a(n) for n = 1..200</a> (terms 1..50 from Andrew Howroyd)
%H Andrew Howroyd, <a href="/A137891/a137891.cs.txt">C# software related to this sequence</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphJoin.html">Graph Join</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HamiltonianPath.html">Hamiltonian Path</a>
%H <a href="/index/Gra#graphs">Index entries for sequences related to graphs, Hamiltonian</a>
%F a(n) = Sum_ { k=1..n } 2*k!*b(n,k)*(k!*b(n,k)+(k-1)!*b(n,k-1)) where b(n,0)=0, b(n,k)=Sum_{ j=1..n-k+1 } j*A130130(j)*A266213(k-1,n-j-k+1) for k>0, n<>2. - _Andrew Howroyd_, Feb 14 2016
%F a(n) ~ c * n!^2, where c = A270047 = 42.12277421168156081166292550105956... . - _Vaclav Kotesovec_, Mar 08 2016
%t b[n_, k_] := If[k == 0, 0, Sum[j*Min[2, j] * Sum[ Binomial[n - j - k, kk - 1]*Binomial[k - 1, kk]*2^kk, {kk, 0, Min[k - 1, n - j - k + 1]}], {j, 1, n - k + 1}]];
%t Flatten[{{2, 24}, Table[Sum[2*k!*b[n, k]*(k!*b[n, k] + (k - 1)!*b[n, k - 1]), {k, 1, n}], {n, 3, 20}]}] (* _Vaclav Kotesovec_, Mar 08 2016, after _Andrew Howroyd_ *)
%Y Cf. A268838 (number of directed Hamiltonian paths in the torus grid graph C_n X C_n).
%Y Cf. A270047.
%K nonn
%O 1,1
%A _Eric W. Weisstein_, Feb 20 2008
%E a(6)-a(7) from _Eric W. Weisstein_, Dec 16 2013
%E a(8)-a(10) from _Eric W. Weisstein_, Dec 24 2013
%E a(1)=2 and a(2)=24 inserted by _Andrew Howroyd_, Feb 14 2016
%E a(11)-a(16) from _Andrew Howroyd_, Feb 14 2016
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