cp's OEIS Frontend

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.

%I A287317 #24 Jan 29 2023 10:45:20
%S A287317 1,10,270,10900,551950,32232060,2070891900,142317232200,
%T A287317 10277494548750,770878551371500,59577647564312020,4717432065143561400,
%U A287317 381091087190569291900,31308955091335405435000,2609450031306515140215000,220199552765301571338488400
%N 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.
%F A287317 a(n) = (2*n)! [x^n] BesselI(0, 2*sqrt(x))^5.
%F A287317 a(n) = binomial(2*n,n)*A169714(n).
%F A287317 a(n) ~ 2^(2*n) * 5^(2*n + 5/2) / (16 * Pi^(5/2) * n^(5/2)). - _Vaclav Kotesovec_, Nov 13 2017
%F A287317 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
%p A287317 A287317_list := proc(len) series(BesselI(0, 2*sqrt(x))^5, x, len);
%p A287317 seq((2*i)!*coeff(%, x, i), i=0..len-1) end: A287317_list(16);
%t A287317 Table[SeriesCoefficient[BesselI[0, 2 Sqrt[x]]^5, {x, 0, n}] (2 n) !, {n, 0, 15}]
%t A287317 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}]
%t A287317 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 *)
%Y A287317 Case k=5 of A287318.
%Y A287317 Cf. A169714, A287316
%Y A287317 1-4 dimensional analogs are A000984, A002894, A002896, A039699.
%K A287317 nonn,easy,walk
%O A287317 0,2
%A A287317 _Peter Luschny_, May 23 2017
%E A287317 Moved original definition to formula section and reworded definition descriptively similar to sequence A039699, by _Dave R.M. Langers_, Oct 12 2022