%I #27 Jun 23 2020 14:31:25

%S 1,4,16,92,432,1884,8582,39736,181936,829672,3793850,17366388,

%T 79441576,363298928,1661695126,7601017276,34767611570,159026305464,

%U 727389859704,3327116203688,15218354613018,69609219627912

%N Number of spanning trees with degrees 1 and 3 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 Vincenzo Librandi, <a href="/A003762/b003762.txt">Table of n, a(n) for n = 1..1000</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 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,30,13,36,48,-76,-14,-36,4,8,-4).

%F Faase gives a 12-term linear recurrence on his web page:

%F a(1) = 1,

%F a(2) = 4,

%F a(3) = 16,

%F a(4) = 92,

%F a(5) = 432,

%F a(6) = 1884,

%F a(7) = 8582,

%F a(8) = 39736,

%F a(9) = 181936,

%F a(10) = 829672,

%F a(11) = 3793850,

%F a(12) = 17366388,

%F a(13) = 79441576,

%F a(14) = 363298928,

%F a(15) = 1661695126 and

%F a(n) = 4a(n-1) - 5a(n-2) + 30a(n-3) + 13a(n-4) + 36a(n-5) + 48a(n-6) - 76a(n-7) - 14a(n-8) - 36a(n-9) + 4a(n-10) + 8a(n-11) - 4a(n-12).

%F G.f.: -x*(8*x^14 +16*x^13 +34*x^12 +12*x^11 +8*x^10 +4*x^9 -4*x^8 +20*x^7 -46*x^6 -48*x^5 -11*x^4 -18*x^3 -5*x^2 -1)/(4*x^12 -8*x^11 -4*x^10 +36*x^9 +14*x^8 +76*x^7 -48*x^6 -36*x^5 -13*x^4 -30*x^3 +5*x^2 -4*x +1). - _Colin Barker_, Sep 07 2012

%t CoefficientList[Series[-(8 * x^14 + 16 * x^13 + 34 * x^12 + 12 * x^11 + 8 * x^10 + 4 * x^9 - 4 * x^8 + 20 * x^7 - 46 * x^6 - 48 * x^5 - 11 * x^4 - 18 * x^3 - 5 * x^2 - 1)/(4 * x^12 - 8 * x^11 - 4 * x^10 + 36 * x^9 + 14 * x^8 + 76 * x^7 - 48 * x^6 - 36 * x^5 - 13 * x^4 - 30 * x^3 + 5 * x^2 - 4 * x + 1), {x, 0, 40}], x] (* _Vincenzo Librandi_, Oct 14 2013 *)

%K nonn,easy

%O 1,2

%A _Frans J. Faase_

%E Added recurrence from Faase's web page. - _N. J. A. Sloane_, Feb 03 2009