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

0,2

Links

Table of n, a(n) for n=0..15.

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

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 *)

Crossrefs

Case k=5 of A287318.

Cf. A169714, A287316

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

Sequence in context: A166811 A089906 A294515 * A084943 A055055 A222998

Adjacent sequences: A287314 A287315 A287316 * A287318 A287319 A287320

Keyword

nonn,easy,walk

Author

Peter Luschny, May 23 2017

Extensions

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

Status

approved