OFFSET
0,3
COMMENTS
Each play (counting untimed downs as part of the previous play) can score at most 8 points for one team.
The same as counting walks that return to the x-axis of x-length n from the origin bounded above by y=8, below by y=-8, and using the steps {[2,2],[3,3],[8,4],[7,5],[6,6],[7,7],[8,8],[2,-2],[3,-3],[8,-4],[7,-5],[6,-6],[7,-7],[8,-8]}.
LINKS
Bryan Ek, Lattice Walk Enumeration, arXiv:1803.10920 [math.CO], 2018.
Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.
FORMULA
G.f.: (16*t^29-16*t^28-56*t^27+52*t^26+100*t^25-52*t^24-136*t^23+108*t^22+66*t^21-71*t^20+134*t^19-5*t^18-320*t^17+50*t^16+78*t^15-47*t^14+60*t^13+78*t^12-158*t^11-8*t^10+31*t^8+t^7+37*t^6-t^5-10*t^4+2*t^3+6*t^2+t-1)/(32*t^33-112*t^32+24*t^31+324*t^30-300*t^29-40*t^28+52*t^27-542*t^26+784*t^25+766*t^24-1610*t^23+166*t^22+792*t^21-563*t^20+420*t^19+681*t^18-1320*t^17+190*t^16+246*t^15-87*t^14+74*t^13+304*t^12-380*t^11+6*t^10-10*t^9+25*t^8-25*t^7+85*t^6-3*t^5-22*t^4+2*t^3+8*t^2+t-1).
EXAMPLE
There is no way to score 1 point so a(1)=0.
The number of ways to be tied at 4-4 is 6: there must be 2 safeties scored by each team which could be ordered in 4 choose 2 ways.
a(5)=24 since there must be 1 safety and 1 field goal for each team and there are 4! ways to order them.
a(n<=8) is fairly easy to compute since the bounds do not come into effect.
MAPLE
taylor((16*t^29-16*t^28-56*t^27+52*t^26+100*t^25-52*t^24-136*t^23+108*t^22+66*t^21-71*t^20+134*t^19-5*t^18-320*t^17+50*t^16+78*t^15-47*t^14+60*t^13+78*t^12-158*t^11-8*t^10+31*t^8+t^7+37*t^6-t^5-10*t^4+2*t^3+6*t^2+t-1)/(32*t^33-112*t^32+24*t^31+324*t^30-300*t^29-40*t^28+52*t^27-542*t^26+784*t^25+766*t^24-1610*t^23+166*t^22+792*t^21-563*t^20+420*t^19+681*t^18-1320*t^17+190*t^16+246*t^15-87*t^14+74*t^13+304*t^12-380*t^11+6*t^10-10*t^9+25*t^8-25*t^7+85*t^6-3*t^5-22*t^4+2*t^3+8*t^2+t-1), t=0, N);
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
Bryan T. Ek, Mar 20 2018
STATUS
approved