%I #39 Aug 23 2023 09:37:24

%S 3,270,20160,1477980,108097935,7903526400,577834413429,42245731959480,

%T 3088601154192960,225808743709815750,16508958287605688193,

%U 1206975861055570636800,88242438021480689844999,6451436286916714206370530,471666820375043557337304000

%N Number of spanning trees in D_4 X P_n.

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

%H Sean A. Irvine, <a href="/A003761/b003761.txt">Table of n, a(n) for n = 1..100</a>

%H F. Faase, <a href="http://www.iwriteiam.nl/Cpaper.zip">On the number of specific spanning subgraphs of the graphs G X P_n</a>, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.

%H F. Faase, <a href="http://www.iwriteiam.nl/counting.html">Counting Hamiltonian cycles in product graphs</a>

%H F. Faase, <a href="http://www.iwriteiam.nl/Cresults.html">Results from the counting program</a>

%H P. Raff, <a href="http://arxiv.org/abs/0809.2551">Spanning Trees in Grid Graphs</a>, arXiv:0809.2551 [math.CO], 2008. [From _Paul Raff_, Mar 06 2009]

%H P. Raff, <a href="http://www.math.rutgers.edu/~praff/span/4/12-13-14-23/index.xml">Analysis of the Number of Spanning Trees of D_4 x P_n</a>. Contains sequence, recurrence, generating function, and more. [From _Paul Raff_, Mar 06 2009]

%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (90,-1313,5850,-9828,5850,-1313,90,-1).

%F a(1) = 3,

%F a(2) = 270,

%F a(3) = 20160,

%F a(4) = 1477980,

%F a(5) = 108097935,

%F a(6) = 7903526400,

%F a(7) = 577834413429,

%F a(8) = 42245731959480 and

%F 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).

%F 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

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

%t 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 *)

%o (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

%K nonn,easy

%O 1,1

%A _Frans J. Faase_

%E Recurrence from Faase's web page added by _N. J. A. Sloane_, Feb 03 2009

%E More terms from _Sean A. Irvine_, Aug 02 2015