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%I #28 Sep 08 2022 08:44:32
%S 45,55125,59719680,64416925125,69471840376125,74922901143552000,
%T 80801651828175064605,87141671714980415665125,
%U 93979154798291442260459520,101353134069755356151903203125
%N Number of spanning trees in W_5 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 P. Raff, <a href="/A003739/b003739.txt">Table of n, a(n) for n = 1..200</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.
%H P. Raff, <a href="http://www.math.rutgers.edu/~praff/span/5/12-13-14-15-23-24-35-45/index.xml">Analysis of the Number of Spanning Trees of W_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/Tra#trees">Index entries for sequences related to trees</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1152,-80640,1442883,-4477824,4477824,-1442883,80640,-1152,1).
%F a(n) = 1152*a(n-1) - 80640*a(n-2) + 1442883*a(n-3) - 4477824*a(n-4) + 4477824*a(n-5) - 1442883*a(n-6) + 80640*a(n-7) - 1152*a(n-8) + a(n-9).
%F G.f.: 45*x*(1 +73*x -3456*x^2 +4534*x^3 +4534*x^4 -3456*x^5 +73*x^6 +x^7)/(1 -1152*x +80640*x^2 -1442883*x^3 +4477824*x^4 -447782*x^5 +1442883*x^6 -80640*x^7 +1152*x^8 -x^9).
%p seq(coeff(series(45*x*(1+73*x-3456*x^2+4534*x^3+4534*x^4-3456*x^5+73*x^6 +x^7)/(1-1152*x+80640*x^2-1442883*x^3+4477824*x^4-447782*x^5+1442883*x^6 -80640*x^7+1152*x^8-x^9), x, n+1), x, n), n = 1..20); # _G. C. Greubel_, Dec 25 2019
%t Rest@CoefficientList[Series[45*x*(1 +73*x -3456*x^2 +4534*x^3 +4534*x^4 -3456*x^5 +73*x^6 +x^7)/(1 -1152*x +80640*x^2 -1442883*x^3 +4477824*x^4 -447782*x^5 +1442883*x^6 -80640*x^7 +1152*x^8 -x^9), {x,0,20}], x] (* _G. C. Greubel_, Dec 25 2019 *)
%o (PARI) my(x='x+O('x^20)); Vec(45*x*(1 +73*x -3456*x^2 +4534*x^3 +4534*x^4 -3456*x^5 +73*x^6 +x^7)/(1 -1152*x +80640*x^2 -1442883*x^3 +4477824*x^4 -447782*x^5 +1442883*x^6 -80640*x^7 +1152*x^8 -x^9)) \\ _G. C. Greubel_, Dec 25 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( 45*x*(1 +73*x -3456*x^2 +4534*x^3 +4534*x^4 -3456*x^5 +73*x^6 +x^7)/(1 -1152*x +80640*x^2 -1442883*x^3 +4477824*x^4 -447782*x^5 +1442883*x^6 -80640*x^7 +1152*x^8 -x^9) )); // _G. C. Greubel_, Dec 25 2019
%o (Sage)
%o def A077952_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( 45*x*(1 +73*x -3456*x^2 +4534*x^3 +4534*x^4 -3456*x^5 +73*x^6 +x^7)/(1 -1152*x +80640*x^2 -1442883*x^3 +4477824*x^4 -447782*x^5 +1442883*x^6 -80640*x^7 +1152*x^8 -x^9) ).list()
%o a=A077952_list(20); a[1:] # _G. C. Greubel_, Dec 25 2019
%K nonn,easy
%O 1,1
%A _Frans J. Faase_
%E Added recurrence from Faase's web page. - _N. J. A. Sloane_, Feb 03 2009
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