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%I #34 Jan 01 2019 06:31:05
%S 4,384,31500,2558976,207746836,16864848000,1369080572444,
%T 111141302329344,9022397309950500,732433860440996736,
%U 59458627396289740076,4826822683620921984000,391839136544897998002484,31809312044806091140235904,2582264604005182130741437500
%N Number of spanning trees in C_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 Alois P. Heinz, <a href="/A003753/b003753.txt">Table of n, a(n) for n = 1..150</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/4/12-13-24-34/index.xml">Analysis of the Number of Spanning Trees of C_4 x P_n</a>. Contains sequence, recurrence, generating function, and more. [Dead link]
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F a(1) = 4,
%F a(2) = 384,
%F a(3) = 31500,
%F a(4) = 2558976,
%F a(5) = 207746836,
%F a(6) = 16864848000 and
%F a(n) = 90a(n-1) - 735a(n-2) + 1548a(n-3) - 735a(n-4) + 90a(n-5) - a(n-6).
%F G.f.: 4x(x^4+6x^3-30x^2+6x+1)/(x^6-90x^5+735x^4-1548x^3+735x^2-90x+1). [_Paul Raff_, Mar 06 2009]
%F a(n) = 4*A001109(n)*A098301(n). [_R. K. Guy_, seqfan list, Mar 28 2009] [From _R. J. Mathar_, Jun 03 2009]
%p a:= n-> (Matrix([[4,0,-4,-384,-31500,-2558976]]). Matrix(6, (i,j)-> if (i=j-1) then 1 elif j=1 then [90, -735, 1548, -735, 90, -1][i] else 0 fi)^(n-1))[1,1]; seq(a(n), n=1..20); # _Alois P. Heinz_, Aug 01 2008
%t a[n_] := (Sqrt[2]/3)*Sinh[n*ArcCosh[3]]*Sinh[n*ArcCosh[7]/2]^2 // Round; Array[a, 20] (* _Jean-François Alcover_, Jan 31 2016 *)
%Y Column k=4 of A173958. - _Alois P. Heinz_, Sep 20 2012
%K nonn
%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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