A043546 Coefficients of asymptotic expansion of return probability for random walk in d-dimensional cubic lattice as a function of d.

0, 1, 2, 7, 35, 215, 1501, 11354, 88978, 675569, 4175664, 1725333, -687775083, -19848956619, -438027976068, -8715988203509, -161989586455204, -2784493824166078, -41530410660307610, -406672888265416456, 4420077014249902362, 456572861717941696791

Offset: 0

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

• sign

Examples

			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 + ...
G.f. = x + 2*x^2 + 7*x^3 + 35*x^4 + 215*x^5 + 1501*x^6 + 11354*x^7 + ...
		

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..60
• Noam D. Elkies, Reply to: Pólya's Random Walk Constants at infinity, MathOverflow Q-83317, Dec 13 2011.
• S. R. Finch, Symmetric Random Walk on n-Dimensional Integer Lattice
• Steven R. Finch, Symmetric Random Walk on n-Dimensional Integer Lattice [Cached copy, with permission of the author]
• P. Flajolet, Reply to: Symmetric random walk on n-dimensional integer lattice, sci.math.research newsgroup posting, 1995.

Programs

Maple

walk:=proc(order) local n,j:
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):
RETURN(asympt(asympt(%,d,2*order+5),d,order+1)):
end:
seq(coeff(convert(walk(20),polynom),d,-n)*2^n,n=0..20); (Ronaldo)

Mathematica

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 *)
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 *)

PARI

N = 20
I1 = sum(n=0,N,x^n/n!^2,O(x^(N+1)));
Iw = subst(I1,x,w^2*x)^(1/(2*w));
g = sum(n=0,N,(2*n)!*polcoeff(Iw,n,x)) + O(w^(N+1));
p = 1 - 1/g
vector(N,n,polcoeff(p,n))
\\ Noam D. Elkies, Dec 13 2011 (see link)

Extensions

Edited by C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 27 2004