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A137891 Number of (directed) Hamiltonian paths in the graph join C_n + C_n of two cycles.

%I A137891 #52 Jan 10 2025 19:25:33
%S A137891 720,13824,383000,14804640,764340024,50913153536,4256161751448,
%T A137891 436618291524000,53955264479804600,7908071556041000064,
%U A137891 1356709951589099693976,269380212536429979520928,61297096735652845698099000,15847986814197933588682229760,4620315237160994963528810238104
%N A137891 Number of (directed) Hamiltonian paths in the graph join C_n + C_n of two cycles.
%H A137891 Vaclav Kotesovec, <a href="/A137891/b137891.txt">Table of n, a(n) for n = 3..200</a> (terms 3..50 from Andrew Howroyd)
%H A137891 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphJoin.html">Graph Join</a>.
%H A137891 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HamiltonianPath.html">Hamiltonian Path</a>.
%H A137891 <a href="/index/Gra#graphs">Index entries for sequences related to graphs, Hamiltonian</a>
%F A137891 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 A137891 a(n) ~ c * n!^2, where c = A270047 = 42.12277421168156081166292550105956... . - _Vaclav Kotesovec_, Mar 08 2016
%t A137891 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 A137891 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_ *)
%o A137891 (PARI) B(n)=polcoef(1/(1 - x*y*(2/(1 - x) - 1)) + O(x*x^n), n)
%o A137891 a(n)={my(v=Vecrev(B(n))); 2*n^2*sum(k=1, n, my(t=v[1+k]*(k-1)!); t*(t + if(k>1, v[k]*(k-2)!)))} \\ _Andrew Howroyd_, Jan 10 2025
%Y A137891 Cf. A130130, A234628, A266213, A270047.
%K A137891 nonn
%O A137891 3,1
%A A137891 _Eric W. Weisstein_, Feb 20 2008
%E A137891 a(6)-a(7) from _Eric W. Weisstein_, Dec 16 2013
%E A137891 a(8)-a(10) from _Eric W. Weisstein_, Dec 24 2013
%E A137891 a(1)=2 and a(2)=24 prepended by _Andrew Howroyd_, Feb 14 2016
%E A137891 a(11)-a(16) from _Andrew Howroyd_, Feb 14 2016
%E A137891 a(1)-a(2) removed by _Andrew Howroyd_, Jan 10 2025