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A043546 Coefficients of asymptotic expansion of return probability for random walk in d-dimensional cubic lattice as a function of d. 1
0, 1, 2, 7, 35, 215, 1501, 11354, 88978, 675569, 4175664, 1725333, -687775083, -19848956619, -438027976068, -8715988203509, -161989586455204, -2784493824166078, -41530410660307610, -406672888265416456, 4420077014249902362, 456572861717941696791 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

P. Flajolet, Reply to: Symmetric random walk on n-dimensional integer lattice, sci.math.research newsgroup posting, 1995.

EXAMPLE

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 + ...

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 *)

PROG

(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).

CROSSREFS

Sequence in context: A214461 A130458 A003575 * A260530 A201690 A080831

Adjacent sequences:  A043543 A043544 A043545 * A043547 A043548 A043549

KEYWORD

sign

AUTHOR

Joe Keane (jgk(AT)jgk.org)

EXTENSIONS

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

STATUS

approved

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Last modified August 31 21:38 EDT 2015. Contains 261257 sequences.