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A054474
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Number of walks on square lattice that start and end at origin after 2n steps, not touching origin at intermediate stages.
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17
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1, 4, 20, 176, 1876, 22064, 275568, 3584064, 47995476, 657037232, 9150655216, 129214858304, 1845409805168, 26606114089024, 386679996988736, 5658611409163008, 83302885723872852, 1232764004638179504, 18327520881735288432, 273595871825723062848
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OFFSET
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0,2
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COMMENTS
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Trajectories returning to the origin are prohibited, contrary to the situation in A094061.
The probability of returning to the origin for the first time after 2n steps is given by a(n)/4^(2*n). If A(x) is a generating function for this sequence, A(x/16) is a generating function for the sequence of probabilities. The sum of these probabilities for n > 0 is 1 unlike in dimensions > 2. - Shel Kaphan, Feb 13 2023
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REFERENCES
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S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 322-331.
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LINKS
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FORMULA
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G.f.: 2 - AGM(1, (1-16*x)^(1/2)).
G.f.: 2 - 1/hypergeom([1/2,1/2],[1],16*x). - Joerg Arndt, Jun 16 2011
Let (in Maple notation) G(x):=4/Pi*EllipticK(4*t)-2/Pi*EllipticF(1/sqrt(2+4*t), 4*t)-2/Pi*EllipticF(1/sqrt(2-4*t), 4*t), then GenFunc(x):=2-1/G(x). - Sergey Perepechko, Sep 11 2004
a(n) ~ Pi * 16^n / (n * log(n)^2) * (1 - (2*gamma + 8*log(2)) / log(n) + (3*gamma^2 + 24*log(2)*gamma + 48*log(2)^2 - Pi^2/2) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 29 2019
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EXAMPLE
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a(5)=22064, i.e., there are 22064 different walks (on a square lattice) that start and end at the origin after 2*5=10 steps, avoiding the origin at intermediate steps.
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MAPLE
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b:= proc(n) b(n):= binomial(2*n, n)^2 end:
a:= proc(n) option remember;
b(n)-add(a(n-i)*b(i), i=1..n-1)
end:
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MATHEMATICA
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m = 18; gf[x_] = 2 - Pi/(2*EllipticK[4*Sqrt[x]]); (List @@ Normal[ Series[ gf[x], {x, 0, m-1}]] /. x -> 1)[[1 ;; m+1]]*Table[4^k, {k, 0, m}] (* Jean-François Alcover, Jun 16 2011, after Vladeta Jovovic *)
CoefficientList[Series[2-Pi/(2*EllipticK[16*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 10 2014 *)
CoefficientList[Series[2-ArithmeticGeometricMean[1, Sqrt[1-16x]], {x, 0, 20}], x] (* Thomas Dybdahl Ahle, Oct 30 2023 *)
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PROG
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(PARI) a(n)=if(n<0, 0, polcoeff(2-agm(1, sqrt(1-16*x+x*O(x^n))), n))
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CROSSREFS
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KEYWORD
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easy,nonn,walk
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AUTHOR
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Alessandro Zinani (alzinani(AT)tin.it), May 19 2000
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STATUS
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approved
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