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%I #89 Feb 06 2024 08:07:23
%S 1,8,168,5120,190120,7939008,357713664,16993726464,839358285480,
%T 42714450658880,2225741588095168,118227198981126144,
%U 6380762273973278464,349019710593278412800,19310744204362333900800,1079054103459778710405120,60818479243449308702049960
%N Number of 4-dimensional cubic lattice walks that start and end at the origin after 2n steps, free to pass through origin at intermediate stages.
%C Generating function G(x) is D-finite with a singular point at x = 1/64 (cf. Graph Link). After summing 300000 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
%C 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 directly from the sequence name, each variable corresponding to a single step in one of the four axis directions. - _Christopher J. Smyth_, Sep 28 2018
%D S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 322-331.
%H Seiichi Manyama, <a href="/A039699/b039699.txt">Table of n, a(n) for n = 0..557</a>
%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>. [broken link]
%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 Bradley Klee, <a href="/A039699/a039699_1.png">Graph of g.f.</a>
%H Gilbert Labelle and Annie Lacasse, <a href="https://doi.org/10.46298/dmtcs.2937">Closed paths whose steps are roots of unity</a>, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.
%H Yen Lee Loh, <a href="https://arxiv.org/abs/1706.03083">A general method for calculating lattice Green functions on the branch cut</a>, arXiv:1706.03083 [math-ph], 2017.
%H J. Novak, <a href="https://arxiv.org/abs/1301.3916">Pólya's random walk theorem</a>, arXiv:1301.3916 [math.PR], 2013.
%F E.g.f.: Sum_{n>=0} a(2*n) * x^(2*n)/(2*n)! = I_0(2*x)^4. (I = Modified Bessel function of the first kind).
%F a(n) = binomial(2*n,n)*A002895(n). - _Mark van Hoeij_, Apr 19 2013
%F a(n) = binomial(2*n,n)^2*hypergeom([1/2,-n,-n,-n],[1,1,1/2-n],1). - _Peter Luschny_, May 23 2017
%F a(n) ~ 2^(6*n+1) / (Pi*n)^2. - _Vaclav Kotesovec_, Nov 13 2017
%F From _Bradley Klee_, Aug 20 2018: (Start)
%F 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
%F 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}}.
%F Sum_{j=0..2,k=0..4} M_{j,k}*a(n-j)*n^k = 0, with
%F M={{0, 0, 0, 0, 1}, {-8, 52, -132, 160, -80}, {768, -3584, 5888, -4096, 1024}}.
%F (End)
%F a(n) = Sum_{i+j+k+l=n, 0<=i,j,k,l<=n} multinomial(2n [i,i,j,j,k,k,l,l]). - _Shel Kaphan_, Jan 16 2023
%e 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.
%p A039699 := n -> binomial(2*n,n)^2*hypergeom([1/2, -n, -n, -n],[1, 1, 1/2 - n], 1):
%p seq(simplify(A039699(n)), n=0..14); # _Peter Luschny_, May 23 2017
%t 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 *)
%t 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 *)
%t 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 *)
%o (PARI)
%o C=binomial;
%o A002895(n) = sum(k=0,n, C(n,k)^2 * C(2*n-2*k,n-k) * C(2*k,k) );
%o a(n)= C(2*n,n) * A002895(n);
%o /* _Joerg Arndt_, Apr 19 2013 */
%Y 1-dimensional, 2-dimensional, 3-dimensional analogs are A000984, A002894, A002896. Pólya Constant: A086232.
%Y Row k=4 of A287318.
%K nonn,nice,easy,walk
%O 0,2
%A Alessandro Zinani (alzinani(AT)tin.it)
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