A054474 Number of walks on square lattice that start and end at origin after 2n steps, not touching origin at intermediate stages.

1, 4, 20, 176, 1876, 22064, 275568, 3584064, 47995476, 657037232, 9150655216, 129214858304, 1845409805168, 26606114089024, 386679996988736, 5658611409163008, 83302885723872852, 1232764004638179504, 18327520881735288432, 273595871825723062848

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.

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

References

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..834

S. R. Finch, Symmetric Random Walk on n-Dimensional Integer Lattice [Broken link]

Steven R. Finch, Symmetric Random Walk on n-Dimensional Integer Lattice [Cached copy, with permission of the author]

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

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

INVERTi transform of A002894. - R. J. Mathar, Sep 24 2020

Example

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.

Maple

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:
seq(a(n), n=0..21);  # Alois P. Heinz, Dec 05 2023

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-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 *)

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, A063887.

Column k=2 of A361397.

Sequence in context: A068965 A185672 A210438 * A364344 A368455 A213144

Adjacent sequences: A054471 A054472 A054473 * A054475 A054476 A054477

Keyword

easy,nonn,walk

Author

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

Status

approved