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A137891 Number of (directed) Hamiltonian paths in the graph join C_n + C_n of two cycles. 4
2, 24, 720, 13824, 383000, 14804640, 764340024, 50913153536, 4256161751448, 436618291524000, 53955264479804600, 7908071556041000064, 1356709951589099693976, 269380212536429979520928, 61297096735652845698099000, 15847986814197933588682229760 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

Andrew Howroyd and Vaclav Kotesovec, Table of n, a(n) for n = 1..200 (terms 1..50 from Andrew Howroyd)

Andrew Howroyd, C# software related to this sequence

Eric Weisstein's World of Mathematics, Graph Join

Eric Weisstein's World of Mathematics, Hamiltonian Path

Index entries for sequences related to graphs, Hamiltonian

FORMULA

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

a(n) ~ c * n!^2, where c = A270047 = 42.12277421168156081166292550105956... . - Vaclav Kotesovec, Mar 08 2016

MATHEMATICA

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}]];

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 *)

CROSSREFS

Cf. A268838 (number of directed Hamiltonian paths in the torus grid graph C_n X C_n).

Cf. A270047.

Sequence in context: A279331 A354378 A119699 * A188959 A093459 A279236

Adjacent sequences:  A137888 A137889 A137890 * A137892 A137893 A137894

KEYWORD

nonn

AUTHOR

Eric W. Weisstein, Feb 20 2008

EXTENSIONS

a(6)-a(7) from Eric W. Weisstein, Dec 16 2013

a(8)-a(10) from Eric W. Weisstein, Dec 24 2013

a(1)=2 and a(2)=24 inserted by Andrew Howroyd, Feb 14 2016

a(11)-a(16) from Andrew Howroyd, Feb 14 2016

STATUS

approved

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Last modified August 12 19:27 EDT 2022. Contains 356077 sequences. (Running on oeis4.)