A033507 Number of matchings in graph P_{4} X P_{n}.

1, 5, 71, 823, 10012, 120465, 1453535, 17525619, 211351945, 2548684656, 30734932553, 370635224561, 4469527322891, 53898461609719, 649966808093412, 7838012982224913, 94519361817920403, 1139818186429110279, 13745178487929574337, 165754445655292452448

Offset: 0

Author

Per H. Lundow

Keywords

• nonn

Examples

			a(1) = 5: the graph is
. o-o-o-o
and the five matchings are
. o o o o
. o-o o o
. o o-o o
. o o o-o
. o-o o-o
		

References

• H. Hosoya and A. Motoyama, An effective algorithm for obtaining polynomials for dimer statistics. Application of operator technique on the topological index to two- and three-dimensional rectangular and torus lattices, J. Math. Phys., 26(1985), 157-167.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500
• David Friedhelm Kind, The Gunport Problem: An Evolutionary Approach, De Montfort University (Leicester, UK, 2020).
• Per Hakan Lundow, Computation of matching polynomials and the number of 1-factors in polygraphs, Research report, No 12, 1996, Department of Math., Umea University, Sweden.
• Per Hakan Lundow, Enumeration of matchings in polygraphs, 1998.
• Index entries for linear recurrences with constant coefficients, signature (9,41,-41,-111,91,29,-23,-1,1).

Crossrefs

Column 4 of triangle A210662. Row sums of A100265.

For perfect matchings see A005178.

Cf. A033508-A033511.

Bisection (even part) gives A260034.

Programs

GAP

a:=[1,5,71,823,10012,120465, 1453535,17525619,211351945];; for n in [10..30] do a[n]:=9*a[n-1]+41*a[n-2]-41*a[n-3]-111*a[n-4]+91*a[n-5] +29*a[n-6]-23*a[n-7]-a[n-8]+a[n-9]; od; a; # G. C. Greubel, Oct 26 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)*(1 -3*x-18*x^2+2*x^3+12*x^4+x^5-x^6)/(1-9*x-41*x^2+41*x^3+111*x^4-91*x^5 -29*x^6+23*x^7+x^8-x^9) )); // G. C. Greubel, Oct 26 2019

Maple

a:=array(0..20,[1, 5, 71, 823, 10012, 120465, 1453535, 17525619, 211351945]):
for j from 9 to 20 do
  a[j]:=9*a[j-1]+41*a[j-2]-41*a[j-3]-111*a[j-4]+91*a[j-5]+
        29*a[j-6]-23*a[j-7]-a[j-8]+a[j-9]
od:
convert(a,list);
# Sergey Perepechko, Apr 24 2013

Mathematica

LinearRecurrence[{9,41,-41,-111,91,29,-23,-1,1},{1,5,71,823,10012,120465, 1453535,17525619,211351945},30] (* Harvey P. Dale, Mar 27 2015 *)

PARI

my(x='x+O('x^30)); Vec((1-x)*(1 -3*x-18*x^2+2*x^3+12*x^4+x^5-x^6)/(1-9*x-41*x^2+41*x^3+111*x^4-91*x^5 -29*x^6+23*x^7+x^8-x^9)) \\ G. C. Greubel, Oct 26 2019

Sage

def A033507_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)*(1 -3*x-18*x^2+2*x^3+12*x^4+x^5-x^6)/(1-9*x-41*x^2+41*x^3+111*x^4-91*x^5 -29*x^6+23*x^7+x^8-x^9) ).list()
A033507_list(30) # G. C. Greubel, Oct 26 2019

Formula

From Sergey Perepechko, Apr 24 2013: (Start)

a(n) = 9*a(n-1) +41*a(n-2) -41*a(n-3) -111*a(n-4) +91*a(n-5) +29*a(n-6) -23*a(n-7) -a(n-8) +a(n-9).

G.f.: (1-x) * (1 -3*x -18*x^2 +2*x^3 +12*x^4 +x^5 -x^6) / (1 -9*x -41*x^2 +41*x^3 +111*x^4 -91*x^5 -29*x^6 +23*x^7 +x^8 -x^9). (End)

Extensions

Edited by N. J. A. Sloane, Nov 15 2009