A003761 Number of spanning trees in D_4 X P_n.

3, 270, 20160, 1477980, 108097935, 7903526400, 577834413429, 42245731959480, 3088601154192960, 225808743709815750, 16508958287605688193, 1206975861055570636800, 88242438021480689844999, 6451436286916714206370530, 471666820375043557337304000

Offset

1,1

References

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

Links

Sean A. Irvine, Table of n, a(n) for n = 1..100

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. [From Paul Raff, Mar 06 2009]

P. Raff, Analysis of the Number of Spanning Trees of D_4 x P_n. Contains sequence, recurrence, generating function, and more. [From Paul Raff, Mar 06 2009]

Index entries for sequences related to trees

Index entries for linear recurrences with constant coefficients, signature (90,-1313,5850,-9828,5850,-1313,90,-1).

Formula

a(1) = 3,

a(2) = 270,

a(3) = 20160,

a(4) = 1477980,

a(5) = 108097935,

a(6) = 7903526400,

a(7) = 577834413429,

a(8) = 42245731959480 and

a(n) = 90*a(n-1) - 1313*a(n-2) + 5850*a(n-3) - 9828*a(n-4) + 5850*a(n-5) - 1313*a(n-6) + 90*a(n-7) - a(n-8).

G.f.: 3*x*(x^6 -67*x^4 +180*x^3 -67*x^2 +1) / (x^8 -90*x^7 +1313*x^6 -5850*x^5 +9828*x^4 -5850*x^3 +1313*x^2 -90*x +1). - Paul Raff, Mar 06 2009

a(n) = 3*A006238(n)*A001109(n). [R. Guy, seqfan list, Mar 28 2009] - R. J. Mathar, Jun 03 2009

Mathematica

CoefficientList[Series[3 (x^6 - 67 x^4 + 180 x^3 - 67 x^2 + 1)/(x^8 - 90 x^7 + 1313 x^6 - 5850 x^5 + 9828 x^4 - 5850 x^3 + 1313 x^2 - 90 x + 1), {x, 0, 33}], x] (* Vincenzo Librandi, Aug 03 2015 *)

Prog

(Magma) I:=[3, 270, 20160, 1477980, 108097935, 7903526400, 577834413429, 42245731959480]; [n le 8 select I[n] else 90*Self(n-1)-1313*Self(n-2)+5850*Self(n-3)-9828*Self(n-4)+5850*Self(n-5)-1313*Self(n-6)+90*Self(n-7)-Self(n-8): n in [1..20]]; // Vincenzo Librandi, Aug 03 2015

Crossrefs

Sequence in context: A319587 A058451 A230373 * A216471 A223037 A171358

Adjacent sequences: A003758 A003759 A003760 * A003762 A003763 A003764

Keyword

nonn,easy

Author

Frans J. Faase

Extensions

Recurrence from Faase's web page added by N. J. A. Sloane, Feb 03 2009

More terms from Sean A. Irvine, Aug 02 2015

Status

approved