A287317 Number of 5-dimensional cubic lattice walks that start and end at origin after 2n steps, free to pass through origin at intermediate stages.

1, 10, 270, 10900, 551950, 32232060, 2070891900, 142317232200, 10277494548750, 770878551371500, 59577647564312020, 4717432065143561400, 381091087190569291900, 31308955091335405435000, 2609450031306515140215000, 220199552765301571338488400

Offset: 0

Author

Peter Luschny, May 23 2017

Keywords

• nonn
• easy
• walk

Crossrefs

Case k=5 of A287318.

Cf. A169714, A287316

1-4 dimensional analogs are A000984, A002894, A002896, A039699.

Programs

Maple

A287317_list := proc(len) series(BesselI(0, 2*sqrt(x))^5, x, len);
seq((2*i)!*coeff(%, x, i), i=0..len-1) end: A287317_list(16);

Mathematica

Table[SeriesCoefficient[BesselI[0, 2 Sqrt[x]]^5, {x, 0, n}] (2 n) !, {n, 0, 15}]
Table[Binomial[2n,n]^2 Sum[(Binomial[n,j]^4/Binomial[2n,2j]) HypergeometricPFQ[{-j,-j,-j}, {1,1/2-j}, 1/4], {j,0,n}], {n,0,15}]
Table[Sum[(2 n)!/(i! j! k! l! (n-i-j-k-l)!)^2, {i,0,n}, {j,0,n-i}, {k,0,n-i-j}, {l,0,n-i-j-k}], {n,0,30}] (* Shel Kaphan, Jan 24 2023 *)

Formula

a(n) = (2*n)! [x^n] BesselI(0, 2*sqrt(x))^5.

a(n) = binomial(2*n,n)*A169714(n).

a(n) ~ 2^(2*n) * 5^(2*n + 5/2) / (16 * Pi^(5/2) * n^(5/2)). - Vaclav Kotesovec, Nov 13 2017

a(n) = Sum_{i+j+k+l+m=n, 0<=i,j,k,l,m<=n} multinomial(2n, [i,i,j,j,k,k,l,l,m,m]). - Shel Kaphan, Jan 24 2023

Extensions

Moved original definition to formula section and reworded definition descriptively similar to sequence A039699, by Dave R.M. Langers, Oct 12 2022