A207817 a(n) = (4*n)! / (n!^4 * (n+1)).

1, 12, 840, 92400, 12612600, 1955457504, 329820499008, 59064793444800, 11062343605599000, 2145275226626532000, 427760079188506384320, 87255985739923260973440, 18139177035549431752363200, 3831766983249199488516960000, 820623729024838763928509760000

Offset: 0

Author

Thotsaporn Thanatipanonda, Feb 20 2012

Keywords

• nonn
• walk

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.

Programs

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 *)

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