|
%I #25 Feb 28 2024 16:13:57
%S 1,189,24576,3046869,375175625,46151368704,5676383392121,
%T 698151521972709,85867005969063936,10560944392853518125,
%U 1298910307853115410641,159755407182415993503744,19648616177810537712940081,2416620034547514872344613709
%N Number of spanning trees in S_5 x P_n.
%H P. Raff, <a href="/A092136/b092136.txt">Table of n, a(n) for n = 1..200</a>
%H F. Faase, <a href="http://www.iwriteiam.nl/counting.html">Counting Hamiltonian cycles in product graphs</a>
%H P. Raff, <a href="http://arxiv.org/abs/0809.2551">Spanning Trees in Grid Graphs</a>, arXiv:0809.2551 [math.CO], 2008.
%H P. Raff, <a href="http://www.math.rutgers.edu/~praff/span/5/12-13-14-15/index.xml">Analysis of the Number of Spanning Trees of S_5 x P_n</a>. Contains sequence, recurrence, generating function, and more.
%H P. Raff, <a href="http://www.myraff.com/projects/spanning-trees-in-grid-graphs">Analysis of the Number of Spanning Trees of Grid Graphs</a>.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (144, -2640, 6930, -2640, 144, -1).
%F a(n) = 144*a(n-1) - 2640*a(n-2) + 6930*a(n-3) - 2640*a(n-4) + 144*a(n-5) - a(n-6). [Modified by _Paul Raff_, Oct 30, 2009]
%F G.f.: -x*(x^4 + 45*x^3 - 45*x-1)/(x^6 - 144*x^5 + 2640*x^4 - 6930*x^3 + 2640*x^2 - 144*x + 1). - Paul Raff (paul(AT)myraff.com), Mar 07 2009
%F a(n) = A004187(n)*(A001906(n))^3 = A004187(n)*A001906(n)*A049684(n). [See R. Guy, seqfan list, Mar 28 2009] - _R. J. Mathar_, Jun 03 2009
%t LinearRecurrence[{7, -1}, {0, 1}, 13] LinearRecurrence[{3, -1}, {0, 1}, 13]^3 // Rest (* _Jean-François Alcover_, Oct 30 2018, after _R. J. Mathar_ *)
%t LinearRecurrence[{144, -2640, 6930, -2640, 144, -1},{1, 189, 24576, 3046869, 375175625, 46151368704}, 14] (* _Ray Chandler_, Feb 28 2024 *)
%Y Cf. A001906, A004187, A049684.
%K nonn
%O 1,2
%A _Ralf Stephan_, Mar 28 2004
|
