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%I #17 Jul 12 2018 00:47:13
%S 1,0,2,2,4,8,14,28,52,78,156,272,520,832,1616,2734,5224,8756,16798,
%T 28192,54118,90644,173876,292816,561574,938748,1802188,3031400,
%U 5812998,9734470,18684588,31367492,60172174,100893834,193598664,324824728,623209036,1045201398,2005438304,3364638978
%N Number of close American football games: number of ways for the game to end after n points have been scored 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 {[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]}.
%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.: -(16*t^58-16*t^57-48*t^56+56*t^55-20*t^54-8*t^53+168*t^52-164*t^51-32*t^50+104*t^49-128*t^48+96*t^47-64*t^46+52*t^45-188*t^44+66*t^43+350*t^42-352*t^41+421*t^40-160*t^39-606*t^38+540*t^37-145*t^36-54*t^35+234*t^34-26*t^33-56*t^32-162*t^31+334*t^30-200*t^29+107*t^28-18*t^27-388*t^26+352*t^25-94*t^24-34*t^23+136*t^22-54*t^21+48*t^20-112*t^19+64*t^18-8*t^17+7*t^16+40*t^15-81*t^14+62*t^13-71*t^12-2*t^11+31*t^10-18*t^9+24*t^8+4*t^7-8*t^6+6*t^5-6*t^4+2*t^3+t^2+1)/(32*t^66-112*t^64+24*t^62+324*t^60-300*t^58-40*t^56+52*t^54-542*t^52+784*t^50+766*t^48-1610*t^46+166*t^44+792*t^42-563*t^40+420*t^38+681*t^36-1320*t^34+190*t^32+246*t^30-87*t^28+74*t^26+304*t^24-380*t^22+6*t^20-10*t^18+25*t^16-25*t^14+85*t^12-3*t^10-22*t^8+2*t^6+8*t^4+t^2-1).
%e There is no way to score 1 point so a(1)=0.
%e There are 2 ways to score 2 or 3 points.
%e a(n<=8) is fairly easy to compute since the bounds do not come into effect.
%e a(9)=78. The unallowable walks are those with 9 points all of the same magnitude: [2,2,2,3],[3,3,3],[2,7],[3,6] (and the negatives and reorderings). A total of 18 unallowable walks. The total walks of length 9 are 2*4*2 (2 and 7 points and ordering) + 2*2*2 (3 and 6) + 2*2*2 (3 and 3 and 3) + 2*2*2*2*4 (2 and 2 and 2 and 3). The total is then 16+8+8+64-18=78.
%p taylor(-(16*t^58-16*t^57-48*t^56+56*t^55-20*t^54-8*t^53+168*t^52-164*t^51-32*t^50+104*t^49-128*t^48+96*t^47-64*t^46+52*t^45-188*t^44+66*t^43+350*t^42-352*t^41+421*t^40-160*t^39-606*t^38+540*t^37-145*t^36-54*t^35+234*t^34-26*t^33-56*t^32-162*t^31+334*t^30-200*t^29+107*t^28-18*t^27-388*t^26+352*t^25-94*t^24-34*t^23+136*t^22-54*t^21+48*t^20-112*t^19+64*t^18-8*t^17+7*t^16+40*t^15-81*t^14+62*t^13-71*t^12-2*t^11+31*t^10-18*t^9+24*t^8+4*t^7-8*t^6+6*t^5-6*t^4+2*t^3+t^2+1)/(32*t^66-112*t^64+24*t^62+324*t^60-300*t^58-40*t^56+52*t^54-542*t^52+784*t^50+766*t^48-1610*t^46+166*t^44+792*t^42-563*t^40+420*t^38+681*t^36-1320*t^34+190*t^32+246*t^30-87*t^28+74*t^26+304*t^24-380*t^22+6*t^20-10*t^18+25*t^16-25*t^14+85*t^12-3*t^10-22*t^8+2*t^6+8*t^4+t^2-1),t=0,N);
%Y Cf. A301379, A301380, A301381.
%K nonn,walk
%O 0,3
%A _Bryan T. Ek_, Mar 20 2018
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