A301381 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.

1, 0, 2, 2, 6, 24, 80, 208, 922, 2310, 8794, 26000, 86632, 274120, 893552, 2837882, 9254642, 29470852, 95567342, 306155908, 987994256, 3174707284, 10228816628, 32893256236, 105937526030, 340778467916, 1097194416030, 3530389210580, 11364292475448, 36571646955122, 117713073900332

Offset: 0

Author

Bryan T. Ek, Mar 20 2018

Keywords

• nonn
• walk

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]}.

Examples

			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.
		

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.

Crossrefs

Cf. A300998, A301379, A301380.

Programs

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);

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).