A378026 Number of simple lattice paths, steps (-1,0,0),(0,-1,0),(0,0,-1), of length 3n from (n,n,n) to the origin, never returning to the diagonal x = y = z before the origin.

1, 6, 54, 816, 14814, 295812, 6262488, 137929392, 3125822238, 72383434332, 1704669773652, 40693683620448, 982302086191752, 23933136140685648, 587728374471479952, 14530886841268923264, 361374588105759096606, 9033515437023805672044, 226844689948433272890396, 5719461854507320708714464

Offset: 0

Author

Markus Kuba, Nov 14 2024

Keywords

• nonn
• walk

Comments

Inversion of A006480 de Bruijn's S(3,n): (3n)!/(n!)^3.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..701
• Markus Kuba and Alois Panholzer, Lattice paths and the diagonal of the cube, arXiv:2411.03930 [math.CO], 2024.

Crossrefs

Cf. A006480, A054474, A000984.

Programs

Maple

b:= proc(n) option remember; (3*n)!/(n!)^3 end:
a:= proc(n) option remember;
      b(n)-add(a(n-i)*b(i), i=1..n-1)
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Nov 15 2024

Mathematica

nmax = 20; CoefficientList[Series[2 - 1/Hypergeometric2F1[1/3, 2/3, 1, 27*x], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 15 2024 *)

Formula

G.f.: 2 - 1/P(z) where P(z) = 2F1(1/3,2/3;1;27z).

INVERTi transform of A006480.

Extensions

More terms from Vaclav Kotesovec, Nov 15 2024