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A054474 Number of walks on square lattice that start and end at origin after 2n steps, not touching origin at intermediate stages. 9
1, 4, 20, 176, 1876, 22064, 275568, 3584064, 47995476, 657037232, 9150655216, 129214858304, 1845409805168, 26606114089024, 386679996988736, 5658611409163008, 83302885723872852, 1232764004638179504, 18327520881735288432 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

1-dimensional and 3-dimensional analogs are A002420 and A049037.

Trajectories returning to the origin are prohibited, contrary to the situation in A094061.

REFERENCES

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 322-331.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..250

S. R. Finch, Symmetric Random Walk on n-Dimensional Integer Lattice

FORMULA

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

G.f.: 2-Pi/(2*EllipticK(4*sqrt(x))). - Vladeta Jovovic, Jun 23 2005

EXAMPLE

a(5)=22064, i.e., there are 22064 different walks (on a square lattice) that start and end at origin after 2*5=10 steps, avoiding origin at intermediate steps.

MATHEMATICA

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-Fran├žois Alcover, Jun 16 2011, after Vladeta Jovovic *)

CoefficientList[Series[2-Pi/(2*EllipticK[16*x]), {x, 0, 20}], x]. - Vaclav Kotesovec, Mar 10 2014

PROG

(PARI) a(n)=if(n<0, 0, polcoeff(2-agm(1, sqrt(1-16*x+x*O(x^n))), n))

CROSSREFS

Cf. A002894, A002420, A049037.

Sequence in context: A068965 A185672 A210438 * A213144 A215873 A066917

Adjacent sequences:  A054471 A054472 A054473 * A054475 A054476 A054477

KEYWORD

easy,nonn,walk

AUTHOR

Alessandro Zinani (alzinani(AT)tin.it), May 19 2000

STATUS

approved

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Last modified March 21 04:59 EDT 2019. Contains 321364 sequences. (Running on oeis4.)