A003739 Number of spanning trees in W_5 X P_n.

45, 55125, 59719680, 64416925125, 69471840376125, 74922901143552000, 80801651828175064605, 87141671714980415665125, 93979154798291442260459520, 101353134069755356151903203125

Offset

1,1

References

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

Links

P. Raff, Table of n, a(n) for n = 1..200

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.

F. Faase, Counting Hamiltonian cycles in product graphs

F. Faase, Results from the counting program

P. Raff, Spanning Trees in Grid Graphs, arXiv:0809.2551 [math.CO], 2008.

P. Raff, Analysis of the Number of Spanning Trees of W_5 x P_n. Contains sequence, recurrence, generating function, and more.

P. Raff, Analysis of the Number of Spanning Trees of Grid Graphs.

Index entries for sequences related to trees

Index entries for linear recurrences with constant coefficients, signature (1152,-80640,1442883,-4477824,4477824,-1442883,80640,-1152,1).

Formula

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).

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).

Maple

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

Mathematica

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 *)

Prog

(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
(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
(Sage)
def A077952_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    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()
a=A077952_list(20); a[1:] # G. C. Greubel, Dec 25 2019

Crossrefs

Sequence in context: A225991 A125113 A354362 * A300198 A145319 A089626

Adjacent sequences: A003736 A003737 A003738 * A003740 A003741 A003742

Keyword

nonn,easy

Author

Frans J. Faase

Extensions

Added recurrence from Faase's web page. - N. J. A. Sloane, Feb 03 2009

Status

approved