A301380 Number of tied close American football games: number of ways for the game to have n scoring plays, never be separated by more than one score after each play, and be tied at the end.

1, 0, 14, 90, 1114, 10718, 113216, 1152540, 11906042, 122269186, 1258639394, 12943924960, 133168371652, 1369830663678, 14091618522696, 144958402357534, 1491181759508514, 15339664777115086, 157798158205312580, 1623258461571800764, 16698349602838663718, 171774768145224952472

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 {[1,8],..,[1,2],[1,-2],..,[1,-8]}.

Examples

			There are no tied games with 1 scoring play. To have tied games after 2 scoring plays requires each team to score the same number of points (7 possibilities) in each play (2 orderings): hence 14 walks.
		

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, A301381.

Programs

Maple

taylor((1-4*t-45*t^2-43*t^3+98*t^4+108*t^5-24*t^6-30*t^7)/(1-4*t-59*t^2-77*t^3+170*t^4+234*t^5-92*t^6-142*t^7-4*t^8+6*t^9),t=0,N);

Formula

G.f.: (1-4*t-45*t^2-43*t^3+98*t^4+108*t^5-24*t^6-30*t^7)/(1-4*t-59*t^2-77*t^3+170*t^4+234*t^5-92*t^6-142*t^7-4*t^8+6*t^9).