A003755 Number of spanning trees in S_4 X P_n.

1, 54, 2240, 89964, 3596725, 143700480, 5740732439, 229334969304, 9161621922880, 365994298083150, 14620972301965259, 584087869159280640, 23333512405041243469, 932141942728566562746, 37237797134599264280000, 1487599121840339002010544, 59427552583207598523644161

Offset: 1

Author

Frans J. Faase

Keywords

• nonn

References

• F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..200
• F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
• F. Faase, Counting Hamiltonian cycles in product graphs
• F. Faase, Results from the counting program
• P. Raff, Spanning Trees in Grid Graphs. arXiv:0809.2551 [math.CO], 2008.
• P. Raff, Analysis of the Number of Spanning Trees of S_4 x P_n. Contains sequence, recurrence, generating function, and more.
• Index entries for sequences related to trees
• Index entries for linear recurrences with constant coefficients, signature (48, -336, 582, -336, 48, -1).

Programs

Maple

a:= n-> (Matrix([[1, 0, -1, -54, -2240, -89964]]). Matrix(6, (i,j)-> if (i=j-1) then 1 elif j=1 then [48,-336,582,-336,48,-1][i] else 0 fi)^(n-1))[1,1]: seq(a(n), n=1..14);  # Alois P. Heinz, Aug 01 2008

Mathematica

LinearRecurrence[{48, -336, 582, -336, 48, -1}, {1, 54, 2240, 89964, 3596725, 143700480}, 17] (* Jean-François Alcover, Aug 06 2018 *)

Formula

a(1) = 1,

a(2) = 54,

a(3) = 2240,

a(4) = 89964,

a(5) = 3596725,

a(6) = 143700480 and

a(n) = 48a(n-1) - 336a(n-2) + 582a(n-3) - 336a(n-4) + 48a(n-5) - a(n-6).

G.f.: x*(x^4+6*x^3-16*x^2+6*x+1)/ ((x^2-6*x+1)*(x^4-42*x^3+83*x^2-42*x+1)). - Paul Raff, Mar 06 2009

a(n) = A001109(n)*A049684(n). [R. Guy, seqfan list, Mar 28 2009] - R. J. Mathar, Jun 03 2009

Extensions

Added recurrence from Faase's web page. - N. J. A. Sloane, Feb 03 2009