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A300391
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The number of paths of length 8*n from the origin to the line y = 3*x/5 with unit east and north steps that stay below the line or touch it.
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1
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1, 7, 525, 58040, 7574994, 1084532963, 164734116407, 26070940600055, 4252443527211637, 709846349042619913, 120679177855928146859, 20822762876863605793639, 3637213213067542990001936, 641912742432770594132245835, 114287840570892852593437353124, 20502971288127330644273350110698
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OFFSET
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0,2
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COMMENTS
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Equivalent to nonnegative walks from (0,0) to (8*n,0) with step set [1,3], [1,-5].
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LINKS
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FORMULA
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G.f. f satisfies f = t^7*f^56 - 2*t^6*f^51 + t^6*f^50 - t^6*f^49 + 7*t^6*f^48 + t^5*f^46 - t^5*f^45 - 3*t^5*f^43 + 5*t^5*f^42 - 6*t^5*f^41 + 21*t^5*f^40 - 3*t^4*f^37 - 3*t^4*f^36 + 8*t^4*f^35 + 10*t^4*f^34 - 15*t^4*f^33 + 35*t^4*f^32 - 2*t^3*f^31 + 2*t^3*f^30 - 9*t^3*f^28 + 22*t^3*f^27 + 10*t^3*f^26 - 20*t^3*f^25 + 35*t^3*f^24 + 3*t^2*f^22 + 5*t^2*f^21 - 9*t^2*f^20 + 18*t^2*f^19 + 5*t^2*f^18 - 15*t^2*f^17 + t*(21*t + 1)*f^16 - t*f^15 + 3*t*f^13 - 3*t*f^12 + 5*t*f^11 + t*f^10 - 6*t*f^9 + 7*t*f^8 + 1.
O.g.f.: A(x) = exp( Sum_{n >= 1} (1/8)*binomial(8*n, 3*n)*x^n/n ) - Bizley.
Recurrence: a(0) = 1 and a(n) = (1/n) * Sum_{k = 0..n-1} (1/8)*binomial(8*n-8*k, 3*n-3*k)*a(k) for n >= 1. (End)
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EXAMPLE
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For n=1, the possible walks are EEEEENNN, EEEENENN, EEEENNEN, EEENEENN, EEENENEN, EENEEENN, EENEENEN.
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CROSSREFS
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KEYWORD
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nonn,walk,easy
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AUTHOR
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STATUS
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approved
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