A003753 Number of spanning trees in C_4 X P_n.

4, 384, 31500, 2558976, 207746836, 16864848000, 1369080572444, 111141302329344, 9022397309950500, 732433860440996736, 59458627396289740076, 4826822683620921984000, 391839136544897998002484, 31809312044806091140235904, 2582264604005182130741437500

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..150
• 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 C_4 x P_n. Contains sequence, recurrence, generating function, and more. [Dead link]
• Index entries for sequences related to trees

Crossrefs

Column k=4 of A173958. - Alois P. Heinz, Sep 20 2012

Programs

Maple

a:= n-> (Matrix([[4,0,-4,-384,-31500,-2558976]]). Matrix(6, (i,j)-> if (i=j-1) then 1 elif j=1 then [90, -735, 1548, -735, 90, -1][i] else 0 fi)^(n-1))[1,1]; seq(a(n), n=1..20);  # Alois P. Heinz, Aug 01 2008

Mathematica

a[n_] := (Sqrt[2]/3)*Sinh[n*ArcCosh[3]]*Sinh[n*ArcCosh[7]/2]^2 // Round; Array[a, 20] (* Jean-François Alcover, Jan 31 2016 *)

Formula

a(1) = 4,

a(2) = 384,

a(3) = 31500,

a(4) = 2558976,

a(5) = 207746836,

a(6) = 16864848000 and

a(n) = 90a(n-1) - 735a(n-2) + 1548a(n-3) - 735a(n-4) + 90a(n-5) - a(n-6).

G.f.: 4x(x^4+6x^3-30x^2+6x+1)/(x^6-90x^5+735x^4-1548x^3+735x^2-90x+1). [Paul Raff, Mar 06 2009]

a(n) = 4*A001109(n)*A098301(n). [R. K. Guy, seqfan list, Mar 28 2009] [From R. J. Mathar, Jun 03 2009]

Extensions

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