|
%I #18 Jul 12 2018 00:47:20
%S 1,14,128,1378,13932,144300,1480376,15245184,156756896,1612836306,
%T 16589928984,170664508406,1755592926518,18059752212038,
%U 185779058543356,1911097952732140,19659326724616886,202234169412143472,2080368880383488938,21400612097499844490,220146623069820835050
%N 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.
%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 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]}.
%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.: (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).
%e 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.
%p 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);
%Y Cf. A300998, A301380, A301381.
%K nonn,walk
%O 0,2
%A _Bryan T. Ek_, Mar 19 2018
|
