A300998 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.

1, 0, 2, 2, 4, 8, 14, 28, 52, 78, 156, 272, 520, 832, 1616, 2734, 5224, 8756, 16798, 28192, 54118, 90644, 173876, 292816, 561574, 938748, 1802188, 3031400, 5812998, 9734470, 18684588, 31367492, 60172174, 100893834, 193598664, 324824728, 623209036, 1045201398, 2005438304, 3364638978

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 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]}.

Examples

			There is no way to score 1 point so a(1)=0.
There are 2 ways to score 2 or 3 points.
a(n<=8) is fairly easy to compute since the bounds do not come into effect.
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.
		

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. A301379, A301380, A301381.

Programs

Maple

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);

Formula

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).