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A039699 Number of 4-dimensional cubic lattice walks that start and end at origin after 2n steps, free to pass through origin at intermediate stages. 5
1, 8, 168, 5120, 190120, 7939008, 357713664, 16993726464, 839358285480, 42714450658880, 2225741588095168, 118227198981126144, 6380762273973278464, 349019710593278412800, 19310744204362333900800 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Generating function G(x) is D-finite with a singular point at x=1/64 (cf. Graph Link). After summing 300K terms, G(1/64)=1.239466... and 1-1/G(1/64)=0.193201... Convergence to A086232 is very slow. - Bradley Klee, Aug 20 2018

a(n) is also the constant term in the expansion of (w + 1/w + x + 1/x + y + 1/y + z + 1/z)^(2n). This follows straight from the sequence name, each variable corresponding to a single step in one of the four axis directions.- Christopher J. Smyth, Sep 28 2018

REFERENCES

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

LINKS

Table of n, a(n) for n=0..14.

S. R. Finch, Symmetric Random Walk on n-Dimensional Integer Lattice [BROKEN LINK]

Bradley Klee, Graph of G.f.

Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjav´k, Iceland DMTCS proc. AO, 2011, 599-610 [Cached copy at the Wayback Machine].

Yen Lee Loh, A general method for calculating lattice Green functions on the branch cut, arXiv:1706.03083 [math-ph], 2017.

J. Novak, Pólya's random walk theorem, arXiv:1301.3916 [math.PR], 2013.

FORMULA

E.g.f.: Sum_{n>=0} a(2*n) * x^(2*n)/(2*n)! = I_0(2*x)^4. (I = Modified Bessel function first kind).

a(n) = binomial(2*n,n)*A002895(n). - Mark van Hoeij, Apr 19 2013

a(n) = binomial(2*n,n)^2*hypergeom([1/2,-n,-n,-n],[1,1,1/2-n],1). - Peter Luschny, May 23 2017

a(n) ~ 2^(6*n+1) / (Pi*n)^2. - Vaclav Kotesovec, Nov 13 2017

From Bradley Klee, Aug 20 2018: (Start)

G.f.: Define G(x)=Sum_{n>=0}a(n)*x^n and G^(j)=(d/dx)^j G(x), then Sum_{j=0..4,k=0..5} M_{j,k}*G^(j)*x^k = 0, with

M={{-8, 768, 0, 0, 0, 0}, {1, -424, 14592, 0, 0, 0}, {0, 7, -1172, 25344, 0, 0}, {0, 0, 6, -640, 10240, 0}, {0, 0, 0, 1, -80, 1024}}.

Sum_{j=0..2,k=0..4} M_{j,k}*a(n-j)*n^k = 0, with

M={{0, 0, 0, 0, 1}, {-8, 52, -132, 160, -80}, {768, -3584, 5888, -4096, 1024}}.

(End)

EXAMPLE

a(5)=7939008, i.e., there are 7939008 different walks that start and end at origin of a 4-dimensional integer lattice after 2*5=10 steps, free to pass through origin at intermediate steps.

MAPLE

A039699 := n -> binomial(2*n, n)^2*hypergeom([1/2, -n, -n, -n], [1, 1, 1/2 - n], 1):

seq(simplify(A039699(n)), n=0..14); # Peter Luschny, May 23 2017

MATHEMATICA

max = 30 (* must be even *); Partition[ CoefficientList[ Series[ BesselI[0, 2 x]^4, {x, 0, max}], x]*Range[0, max]!, 2][[All, 1]] (* Jean-François Alcover, Oct 05 2011 *)

With[{nn=30}, Take[CoefficientList[Series[BesselI[0, 2x]^4, {x, 0, nn}], x] Range[0, nn]!, {1, -1, 2}]] (* Harvey P. Dale, Aug 09 2013 *)

RecurrenceTable[{256*(n-1)^2*(2*n-3)*(2*n-1)*a[n-2] - 4*(2*n-1)^2*(5*n^2-5*n+2)*a[n-1] + n^4*a[n]==0, a[0]==1, a[1]==8}, a, {n, 0, 100}] (* Bradley Klee, Aug 20 2018 *)

PROG

(PARI)

C=binomial;

A002895(n) = sum(k=0, n, C(n, k)^2 * C(2*n-2*k, n-k) * C(2*k, k) );

a(n)= C(2*n, n) * A002895(n);

/* Joerg Arndt, Apr 19 2013 */

CROSSREFS

1-dimensional, 2-dimensional, 3-dimensional analogs are A000984, A002894, A002896. Pólya Constant: A086232.

Sequence in context: A090228 A220808 A221022 * A253974 A254459 A254129

Adjacent sequences:  A039696 A039697 A039698 * A039700 A039701 A039702

KEYWORD

nonn,nice,easy,walk

AUTHOR

Alessandro Zinani (alzinani(AT)tin.it)

STATUS

approved

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Last modified November 18 21:04 EST 2018. Contains 317331 sequences. (Running on oeis4.)