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A301379
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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.
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3
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1, 14, 128, 1378, 13932, 144300, 1480376, 15245184, 156756896, 1612836306, 16589928984, 170664508406, 1755592926518, 18059752212038, 185779058543356, 1911097952732140, 19659326724616886, 202234169412143472, 2080368880383488938, 21400612097499844490, 220146623069820835050
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OFFSET
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0,2
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COMMENTS
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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]}.
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LINKS
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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.
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FORMULA
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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).
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EXAMPLE
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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.
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MAPLE
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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);
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CROSSREFS
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Cf. A300998, A301380, A301381.
Sequence in context: A166794 A229386 A208047 * A222571 A038841 A240189
Adjacent sequences: A301376 A301377 A301378 * A301380 A301381 A301382
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KEYWORD
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nonn,walk
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AUTHOR
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Bryan T. Ek, Mar 19 2018
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STATUS
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approved
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