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A026325
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Number of paths in the plane x >= 0 and y >= -2, from (0,0) to (n,0), and consisting of steps U = (1,1), D = (1,-1) and H = (1,0).
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3
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1, 1, 3, 7, 19, 51, 140, 386, 1071, 2983, 8338, 23376, 65715, 185199, 523134, 1480872, 4200411, 11936619, 33981063, 96897759, 276739029, 791532973, 2267119660, 6502108902, 18671460905, 53680763201, 154507444731, 445190930863, 1284064525987
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OFFSET
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0,3
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COMMENTS
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Previous name: Number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 2, s(n) = 2. Also T(n,n), where T is the array in A026323.
Number of paths in the plane x >= 0 and y >= -2, from (0,0) to (n,0), and consisting of steps U = (1,1), D = (1,-1) and H = (1,0). For example, for n=3, we have the 7 paths: HHH, UDH, HUD, UHD, HDU, DUH, DHU. - José Luis Ramírez Ramírez, Apr 20 2015
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LINKS
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FORMULA
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(n + 6)*a(n) + (-4*n - 15)*a(n-1) + (n - 3)*a(n-2) + 6*n*a(n-3) = 0. - R. J. Mathar, Jul 23 2017
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MAPLE
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gf := sqrt(4 - 8*x - 12*x^2)*(1/x^5 - 1/x^4 - 1/(4*x^6)):
ser := series(gf, x, 36): seq(coeff(ser, x, n), n=0..28);
A026325 := proc(n) option remember; ifelse(n < 3, [1, 1, 3][n + 1],
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MATHEMATICA
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CoefficientList[Series[1/(1 - x - x^2 ((1 - x - (1 - 2 x - 3 x^2)^(1/2))/(2 x^2) + 1/(1 - x - x^2 / (1 - x)))), {x, 0, 40}], x] (* Vincenzo Librandi, Apr 21 2015 *)
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PROG
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(PARI) x='x+O('x^50); Vec(1/(1 - x - x^2*((1 - x - (1 - 2*x - 3*x^2)^(1/2))/(2*x^2) + 1/(1 - x - x^2/(1 - x))))) \\ G. C. Greubel, Feb 15 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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