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

0,2

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.

Formula

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

INVERTi transform of A006480.

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 *)

Crossrefs

Cf. A006480, A054474, A000984.

Sequence in context: A387976 A360545 A217238 * A171681 A397882 A267837

Adjacent sequences: A378023 A378024 A378025 * A378027 A378028 A378029

Keyword

nonn,walk

Author

Markus Kuba, Nov 14 2024

Extensions

More terms from Vaclav Kotesovec, Nov 15 2024

Status

approved