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Number of three-dimensional random walks with 2n steps in the wedge region x >= y >= z, beginning and ending at the origin without crossing the wedge boundary.
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%I #19 Feb 16 2025 08:33:23

%S 1,2,12,120,1610,25956,474012,9475752,202921290,4587734580,

%T 108376022040,2654745191280,67043341981980,1737717447946200,

%U 46062204663294000,1245096242017227360,34239776369652506970,956050033694583839220

%N Number of three-dimensional random walks with 2n steps in the wedge region x >= y >= z, beginning and ending at the origin without crossing the wedge boundary.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.9 Polya's random walk constants, p. 326.

%H Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/PolyasRandomWalkConstants.html">Polya's Random Walk Constants</a>

%F a(n) = sum_{k=0..n} (2n)!*(2k)!/((n-k)!*(n+1-k)!*k!^2*(k+1)!^2).

%F a(n) = C(n) * 3F2(1/2, -n-1, -n; 2, 2; 4) where C(n) is the n-th Catalan number and 3F2 the hypergeometric function.

%F a(n) ~ 2^(2*n-4) * 3^(2*n+9/2) / (Pi^(3/2) * n^(9/2)). - _Vaclav Kotesovec_, Nov 13 2014

%F Recurrence: n*(n+2)^2*a(n) = 2*(2*n-1)*(10*n^2 + 2*n - 3)*a(n-1) - 36*(n-1)*(2*n-3)*(2*n-1)*a(n-2). - _Vaclav Kotesovec_, May 14 2016

%e For 2n=4, the 12 acceptable walks are:

%e (0, 0, -1), (0, -1, -1), (0, 0, -1), (0 ,0, 0);

%e (0, 0, -1), (0, 0, 0), (0, 0, -1), (0 ,0, 0);

%e (0, 0, -1), (0, 0, 0), (1, 0, 0), (0 ,0, 0);

%e (0, 0, -1), (1, 0, -1), (0, 0, -1), (0 ,0, 0);

%e (0, 0, -1), (1, 0, -1), (1, 0, 0), (0 ,0, 0);

%e (1, 0, 0), (0, -1, -1), (0, 0, -1), (0 ,0, 0);

%e (1, 0, 0), (0, 0, 0), (0, 0, -1), (0 ,0, 0);

%e (1, 0, 0), (0, 0, 0), (1, 0, 0), (0 ,0, 0);

%e (1, 0, 0), (1, 0, -1), (0, 0, -1), (0 ,0, 0);

%e (1, 0, 0), (1, 0, -1), (1, 0, 0), (0 ,0, 0);

%e (1, 0, 0), (1, 1, 0), (0, 0, -1), (0 ,0, 0);

%e (1, 0, 0), (1, 1, 0), (1, 0, 0), (0 ,0, 0).

%t a[n_] := CatalanNumber[n]*HypergeometricPFQ[{1/2, -n-1, -n}, {2, 2}, 4]; Table[a[n], {n, 0, 20}]

%Y Cf. A000108, A002896, A086230, A086231.

%K nonn,walk,easy,changed

%O 0,2

%A _Jean-François Alcover_, Nov 12 2014