login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 5 09:15 EST 2016. Contains 278762 sequences.