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Number of tied close American football games: number of ways for the game to end at the score of n to n and never be separated by more than one score after each play.
3

%I #16 Jul 12 2018 00:47:36

%S 1,0,2,2,6,24,80,208,922,2310,8794,26000,86632,274120,893552,2837882,

%T 9254642,29470852,95567342,306155908,987994256,3174707284,10228816628,

%U 32893256236,105937526030,340778467916,1097194416030,3530389210580,11364292475448,36571646955122,117713073900332

%N Number of tied close American football games: number of ways for the game to end at the score of n to n and never be separated by more than one score after each play.

%C Each play (counting untimed downs as part of the previous play) can score at most 8 points for one team.

%C 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]}.

%H Bryan Ek, <a href="https://arxiv.org/abs/1803.10920">Lattice Walk Enumeration</a>, arXiv:1803.10920 [math.CO], 2018.

%H Bryan Ek, <a href="https://arxiv.org/abs/1804.05933">Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics</a>, arXiv:1804.05933 [math.CO], 2018.

%F 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).

%e There is no way to score 1 point so a(1)=0.

%e 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.

%e a(5)=24 since there must be 1 safety and 1 field goal for each team and there are 4! ways to order them.

%e a(n<=8) is fairly easy to compute since the bounds do not come into effect.

%p 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);

%Y Cf. A300998, A301379, A301380.

%K nonn,walk

%O 0,3

%A _Bryan T. Ek_, Mar 20 2018