A301379 Number of close American football games: number of ways for the game to have n scoring plays and never be separated by more than one score after each play.

1, 14, 128, 1378, 13932, 144300, 1480376, 15245184, 156756896, 1612836306, 16589928984, 170664508406, 1755592926518, 18059752212038, 185779058543356, 1911097952732140, 19659326724616886, 202234169412143472, 2080368880383488938, 21400612097499844490, 220146623069820835050

Offset

0,2

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 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..20.

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+10*t+13*t^2-37*t^3-40*t^4+28*t^5+26*t^6-2*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

For n=1, any step is valid. For n=2, any walk with steps of opposite direction is valid while [[1,3],[1,6]] is an example of an invalid walk.

Maple

taylor((1+10*t+13*t^2-37*t^3-40*t^4+28*t^5+26*t^6-2*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, A301380, A301381.

Sequence in context: A166794 A229386 A208047 * A222571 A038841 A240189

Adjacent sequences: A301376 A301377 A301378 * A301380 A301381 A301382

Keyword

nonn,walk

Author

Bryan T. Ek, Mar 19 2018

Status

approved