OFFSET
0,2
COMMENTS
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.
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).
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.
FORMULA
G.f.: 3F2(1/4,1/2,3/4;1,2;256*x). - Benedict W. J. Irwin, Jul 13 2016
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
MAPLE
with(combinat, multinomial): seq(multinomial(4*n, n$4)/(n+1), n=0..20);
MATHEMATICA
CoefficientList[Series[HypergeometricPFQ[{1/4, 1/2, 3/4}, {1, 2}, 256 x], {x, 0, 20}], x] (* Benedict W. J. Irwin, Jul 13 2016 *)
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
Thotsaporn Thanatipanonda, Feb 20 2012
STATUS
approved