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%I #39 Apr 14 2019 10:12:58
%S 0,1,2,7,35,215,1501,11354,88978,675569,4175664,1725333,-687775083,
%T -19848956619,-438027976068,-8715988203509,-161989586455204,
%U -2784493824166078,-41530410660307610,-406672888265416456,4420077014249902362,456572861717941696791
%N Coefficients of asymptotic expansion of return probability for random walk in d-dimensional cubic lattice as a function of d.
%D S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 322-331.
%H Vincenzo Librandi, <a href="/A043546/b043546.txt">Table of n, a(n) for n = 0..60</a>
%H Noam D. Elkies, Reply to: <a href="http://mathoverflow.net/questions/83317/polyas-random-walk-constants-at-infinity">Pólya's Random Walk Constants at infinity</a>, MathOverflow Q-83317, Dec 13 2011.
%H S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/polya/flajolet.html">Symmetric Random Walk on n-Dimensional Integer Lattice</a>
%H Steven R. Finch, <a href="/A054474/a054474.txt">Symmetric Random Walk on n-Dimensional Integer Lattice</a> [Cached copy, with permission of the author]
%H P. Flajolet, <a href="http://mathforum.org/epigone/sci.math.research/nogimflon">Reply to: Symmetric random walk on n-dimensional integer lattice</a>, sci.math.research newsgroup posting, 1995.
%e Flajolet's code gives the following asymptotic expansion: p(d) = 1/(2*d) + 2/(2*d)^2 + 7/(2*d)^3 + 35/(2*d)^4 + 215/(2*d)^5 + 1501/(2*d)^6 + 11354/(2*d)^7 + 88978/(2*d)^8 + 675569/(2*d)^9 + 4175664/(2*d)^10 + 1725333/(2*d)^11 - 687775083/(2*d)^12 - 19848956619/(2*d)^13 - 438027976068/(2*d)^14 - 8715988203509/(2*d)^15 - 161989586455204/(2*d)^16 - 2784493824166078/(2*d)^17 - 41530410660307610/(2*d)^18 - 406672888265416456/(2*d)^19 + 4420077014249902362/(2*d)^20 + ...
%e G.f. = x + 2*x^2 + 7*x^3 + 35*x^4 + 215*x^5 + 1501*x^6 + 11354*x^7 + ...
%p walk:=proc(order) local n,j:
%p j:=sum(t^(2*n)/(2*d)^(2*n)/n!^2,n=0..order): eval(subs(O=0, asympt(exp(d*log(j)),d,order+2)))*exp(-t): 1-1/int(%,t=0..infinity):
%p RETURN(asympt(asympt(%,d,2*order+5),d,order+1)):
%p end:
%p seq(coeff(convert(walk(20),polynom),d,-n)*2^n,n=0..20); (Ronaldo)
%t nn = 20; I1 = Sum[x^n/n!^2, {n, 0, nn}]; Iw = (I1 /. x -> w^2*x)^(1/(2*w)); g = Sum[(2*n)!*SeriesCoefficient[Iw, {x, 0, n}], {n, 0, nn}]; p = 1 - 1/g; Table[SeriesCoefficient[p, {w, 0, n}], {n, 0, nn}] (* _Jean-François Alcover_, Jan 08 2014, after the PARI code by _Noam D. Elkies_ *)
%t a[ n_] := If[n < 0, 0, With[{A = Series[ BesselI[ 0, 2 Sqrt[y y x]]^(1/(2 y)), {x, 0, n}]}, SeriesCoefficient[ 1 - 1 / (Sum[(2 k)! SeriesCoefficient[ A, {x, 0, k}], {k, 0, n}]), {y, 0, n}]]]; (* _Michael Somos_, May 25 2014 *)
%o (PARI)
%o N = 20
%o I1 = sum(n=0,N,x^n/n!^2,O(x^(N+1)));
%o Iw = subst(I1,x,w^2*x)^(1/(2*w));
%o g = sum(n=0,N,(2*n)!*polcoeff(Iw,n,x)) + O(w^(N+1));
%o p = 1 - 1/g
%o vector(N,n,polcoeff(p,n))
%o \\ _Noam D. Elkies_, Dec 13 2011 (see link)
%K sign
%O 0,3
%A Joe Keane (jgk(AT)jgk.org)
%E Edited by C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 27 2004
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