A086230 -id:A086230 - cp's OEIS frontend

A242815 Decimal expansion of the expected number of returns to the origin of a random walk on a 7-d lattice.

1, 0, 9, 3, 9, 0, 6, 3, 1, 5, 5, 8, 7, 8, 4, 7, 9, 9, 6, 6, 8, 3, 2, 7, 1, 8, 2, 3, 5, 5, 9, 0, 1, 9, 8, 6, 3, 7, 1, 1, 2, 8, 9, 9, 7, 7, 1, 6, 4, 9, 6, 1, 1, 5, 4, 4, 9, 1, 6, 8, 9, 0, 7, 3, 8, 8, 6, 1, 2, 6, 5, 4, 5, 7, 0, 5, 0, 8, 0, 5, 2, 2, 8, 4, 4, 8, 9, 5, 1, 9, 1, 9, 7, 2, 9, 8, 5, 5, 9, 8, 7, 5, 7, 2, 9, 9

Offset: 1

Jean-François Alcover, May 23 2014

			1.09390631558784799668327...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.9 Polya's random walk constants, p. 323.

Marc Mezzarobba, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pólya's Random Walk Constants.

Cf. A086230, A086231, A086232, A086233, A086234, A086235, A086236, A242812, A242813, A242814, A242816.

m7:= int(exp(-t)*BesselI(0, t/7)^7, t=0..infinity):
s:= convert(evalf(m7, 120), string):
map(parse, subs("."=NULL, [seq(i, i=s)]))[]; # Alois P. Heinz, May 23 2014

d = 7; d/Pi^d*NIntegrate[(d - Sum[Cos[t[k]], {k, 1, d}])^-1, Sequence @@ Table[{t[k], 0, Pi}, {k, 1, d}] // Evaluate] // RealDigits[#, 10, 7]& // First

m(d) = d/(2*Pi)^d*multipleIntegral(-Pi..Pi) (d-sum_(k=1..d) cos(t_k))^(-1) dt_1 dt_2 ... dt_d, where d is the lattice dimension.
m(d) = Integral_{t>0} exp(-t)*BesselI(0,t/d)^d dt where BesselI(0,x) is the zeroth modified Bessel function.
Equals 1/(1 - A086235). - Amiram Eldar, Aug 28 2020

More terms from Alois P. Heinz, May 23 2014

A242761 Decimal expansion of the escape probability for a random walk on the 3-D cubic lattice (a Polya random walk constant).

Original entry on oeis.org

6, 5, 9, 4, 6, 2, 6, 7, 0, 4, 4, 9, 0, 0, 0, 8, 5, 7, 1, 7, 3, 7, 2, 6, 8, 1, 5, 5, 6, 7, 0, 9, 7, 1, 0, 3, 2, 8, 9, 3, 9, 1, 7, 8, 2, 8, 7, 5, 6, 9, 7, 9, 0, 2, 2, 3, 6, 7, 6, 3, 8, 9, 4, 6, 2, 2, 2, 0, 8, 0, 3, 0, 5, 4, 1, 0, 3, 7, 6, 1, 5, 3, 5, 7, 4, 7, 1, 9, 1, 8, 1, 1, 0, 9, 4, 2, 8, 6, 9, 0

Offset: 0

Jean-François Alcover, May 22 2014

			0.6594626704490008571737268155670971...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.9, p. 322.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's MathWorld, Polya's Random Walk Constants

Cf. A086230, A086231, A086232-A086236, A043546, A293237, A293238, A242812-A242816.

SetDefaultRealField(RealField(100)); R:= RealField(); (16*Sqrt(2/3)*Pi(R)^3)/(Gamma(1/24)*Gamma(5/24)*Gamma(7/24)*Gamma(11/24)); // G. C. Greubel, Oct 26 2018

p = (16*Sqrt[2/3]*Pi^3)/(Gamma[1/24]*Gamma[5/24]*Gamma[7/24]*Gamma[11/24]); RealDigits[p, 10, 100] // First

default(realprecision, 100); (16*sqrt(2/3)*Pi^3)/(gamma(1/24)* gamma(5/24)*gamma(7/24)*gamma(11/24)) \\ G. C. Greubel, Oct 26 2018

Equals (16*sqrt(2/3)*Pi^3)/(Gamma(1/24)*Gamma(5/24)*Gamma(7/24)*Gamma(11/24)), where Gamma is the Euler Gamma function.

A245067 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.

Original entry on oeis.org

1, 2, 12, 120, 1610, 25956, 474012, 9475752, 202921290, 4587734580, 108376022040, 2654745191280, 67043341981980, 1737717447946200, 46062204663294000, 1245096242017227360, 34239776369652506970, 956050033694583839220

Offset: 0

Jean-François Alcover, Nov 12 2014

			For 2n=4, the 12 acceptable walks are:
(0, 0, -1), (0, -1, -1), (0, 0, -1), (0 ,0, 0);
(0, 0, -1), (0, 0, 0), (0, 0, -1), (0 ,0, 0);
(0, 0, -1), (0, 0, 0), (1, 0, 0), (0 ,0, 0);
(0, 0, -1), (1, 0, -1), (0, 0, -1), (0 ,0, 0);
(0, 0, -1), (1, 0, -1), (1, 0, 0), (0 ,0, 0);
(1, 0, 0), (0, -1, -1), (0, 0, -1), (0 ,0, 0);
(1, 0, 0), (0, 0, 0), (0, 0, -1), (0 ,0, 0);
(1, 0, 0), (0, 0, 0), (1, 0, 0), (0 ,0, 0);
(1, 0, 0), (1, 0, -1), (0, 0, -1), (0 ,0, 0);
(1, 0, 0), (1, 0, -1), (1, 0, 0), (0 ,0, 0);
(1, 0, 0), (1, 1, 0), (0, 0, -1), (0 ,0, 0);
(1, 0, 0), (1, 1, 0), (1, 0, 0), (0 ,0, 0).

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

Eric Weisstein's MathWorld, Polya's Random Walk Constants

Cf. A000108, A002896, A086230, A086231.

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

a(n) = sum_{k=0..n} (2n)!*(2k)!/((n-k)!*(n+1-k)!*k!^2*(k+1)!^2).
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.
a(n) ~ 2^(2*n-4) * 3^(2*n+9/2) / (Pi^(3/2) * n^(9/2)). - Vaclav Kotesovec, Nov 13 2014
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

cp's OEIS Frontend

A242815 Decimal expansion of the expected number of returns to the origin of a random walk on a 7-d lattice.

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A242761 Decimal expansion of the escape probability for a random walk on the 3-D cubic lattice (a Polya random walk constant).

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A245067 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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