%I A054474
%S A054474 1,4,20,176,1876,22064,275568,3584064,47995476,657037232,9150655216,
%T A054474 129214858304,1845409805168,26606114089024,386679996988736,
%U A054474 5658611409163008,83302885723872852,1232764004638179504,18327520881735288432
%N A054474 Number of walks on square lattice that start and end at origin after 
               2n steps, not touching origin at intermediate stages.
%C A054474 1-dimensional and 3-dimensional analogues are A002420 and A049037.
%C A054474 Trajectories returning to the origin are prohibited, contrary to the 
               situation in A094061.
%D A054474 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 322-331.
%H A054474 S. R. Finch, <a href="http://algo.inria.fr/bsolve/constant/polya/flajolet.html">
               Symmetric Random Walk on n-Dimensional Integer Lattice</a>
%F A054474 G.f.: 2-AGM(1, (1-16x)^(1/2)).
%F A054474 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 (persn(AT)aport.ru), Sep 11 2004
%F A054474 G.f.: 2-Pi/(2*EllipticK(4*sqrt(x))). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Jun 23 2005
%e A054474 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.
%o A054474 (PARI) a(n)=if(n<0,0,polcoeff(2-agm(1,sqrt(1-16*x+x*O(x^n))),n))
%Y A054474 Cf. A002894, A002420, A049037.
%Y A054474 Sequence in context: A065526 A032333 A068965 this_sequence A066917 A032081 
               A034235
%Y A054474 Adjacent sequences: A054471 A054472 A054473 this_sequence A054475 A054476 
               A054477
%K A054474 easy,nonn
%O A054474 0,2
%A A054474 Alessandro Zinani (alzinani(AT)tin.it), May 19 2000