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A277360
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Number of self-avoiding planar walks starting at (0,0), ending at (n,n), remaining in the first quadrant and using steps (0,1), (1,0), (1,1), (-1,1), and (1,-1).
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3
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1, 9, 491, 64159, 15314361, 5799651689, 3193954129651, 2410542221526399, 2388182999073694001, 3006071549433968619529, 4685653563347872021885371, 8859314350383162594502273439, 19975392290718104323103596377961, 52949467092712165429316121638458089
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = (16*n^2-4*n-1)*a(n-1) - n*(4*n-6)*a(n-2) for n>1, a(0)=1, a(1)=9.
a(n) = (2n)! * [x^(2n)] exp(-x/2)/(1-2*x)^(5/4).
a(n) ~ sqrt(Pi) * 2^(4*n + 13/4) * n^(2*n + 3/4) / (Gamma(1/4) * exp(2*n + 1/4)). - Vaclav Kotesovec, Oct 13 2016
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MAPLE
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a:= proc(n) option remember; `if`(n<2, 8*n+1,
(16*n^2-4*n-1)*a(n-1)-n*(4*n-6)*a(n-2))
end:
seq(a(n), n=0..15);
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MATHEMATICA
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a[n_] := a[n] = If[n<2, 8n+1, (16n^2 - 4n - 1) a[n-1] - n (4n-6) a[n-2]];
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CROSSREFS
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KEYWORD
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nonn,walk
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AUTHOR
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STATUS
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approved
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