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%I #24 Sep 27 2020 15:21:00
%S 1,12,840,92400,12612600,1955457504,329820499008,59064793444800,
%T 11062343605599000,2145275226626532000,427760079188506384320,
%U 87255985739923260973440,18139177035549431752363200,3831766983249199488516960000,820623729024838763928509760000
%N a(n) = (4*n)! / (n!^4 * (n+1)).
%C Number of walks in 4-dimensions using steps (1,0,0,0), (0,1,0,0), (0,0,1,0) and (0,0,0,1) from (0,0,0,0) to (n,n,n,n) such that after each step we have y>=x.
%C Number of possible necklaces consisting of n white beads, n-1 red beads, n-1 green beads, and n-1 blue beads (two necklaces are considered equivalent if they differ by a cyclic permutation).
%C Note: the generalizations of this formula and the relation between d-dimensional walks and d-colored necklaces are also true for all d, d>=5.
%F G.f.: 3F2(1/4,1/2,3/4;1,2;256*x). - _Benedict W. J. Irwin_, Jul 13 2016
%F D-finite with recurrence n^2*(n+1)*a(n) -8*(4*n-3)*(2*n-1)*(4*n-1)*a(n-1)=0. - _R. J. Mathar_, Sep 27 2020
%p with(combinat, multinomial): seq(multinomial(4*n, n$4)/(n+1), n=0..20);
%t CoefficientList[Series[HypergeometricPFQ[{1/4, 1/2, 3/4}, {1, 2}, 256 x], {x, 0, 20}], x] (* _Benedict W. J. Irwin_, Jul 13 2016 *)
%K nonn,walk
%O 0,2
%A _Thotsaporn Thanatipanonda_, Feb 20 2012
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