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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.
6
1, 10, 270, 10900, 551950, 32232060, 2070891900, 142317232200, 10277494548750, 770878551371500, 59577647564312020, 4717432065143561400, 381091087190569291900, 31308955091335405435000, 2609450031306515140215000, 220199552765301571338488400
OFFSET
0,2
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.
1-4 dimensional analogs are A000984, A002894, A002896, A039699.
Sequence in context: A166811 A089906 A294515 * A084943 A055055 A222998
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