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

0,3

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

Links

Table of n, a(n) for n=0..21.

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.: (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).

Example

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.

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);

Crossrefs

Cf. A300998, A301379, A301381.

Sequence in context: A054487 A200191 A266805 * A047639 A202291 A010930

Adjacent sequences: A301377 A301378 A301379 * A301381 A301382 A301383

Keyword

nonn,walk

Author

Bryan T. Ek, Mar 20 2018

Status

approved