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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.

%I A378026 #27 Apr 04 2025 15:02:28
%S A378026 1,6,54,816,14814,295812,6262488,137929392,3125822238,72383434332,
%T A378026 1704669773652,40693683620448,982302086191752,23933136140685648,
%U A378026 587728374471479952,14530886841268923264,361374588105759096606,9033515437023805672044,226844689948433272890396,5719461854507320708714464
%N 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.
%C A378026 Inversion of A006480 de Bruijn's S(3,n): (3n)!/(n!)^3.
%H A378026 Alois P. Heinz, <a href="/A378026/b378026.txt">Table of n, a(n) for n = 0..701</a>
%H A378026 Markus Kuba and Alois Panholzer, <a href="https://arxiv.org/abs/2411.03930">Lattice paths and the diagonal of the cube</a>, arXiv:2411.03930 [math.CO], 2024.
%F A378026 G.f.: 2 - 1/P(z) where P(z) = 2F1(1/3,2/3;1;27z).
%F A378026 INVERTi transform of A006480.
%p A378026 b:= proc(n) option remember; (3*n)!/(n!)^3 end:
%p A378026 a:= proc(n) option remember;
%p A378026       b(n)-add(a(n-i)*b(i), i=1..n-1)
%p A378026     end:
%p A378026 seq(a(n), n=0..22);  # _Alois P. Heinz_, Nov 15 2024
%t A378026 nmax = 20; CoefficientList[Series[2 - 1/Hypergeometric2F1[1/3, 2/3, 1, 27*x], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 15 2024 *)
%Y A378026 Cf. A006480, A054474, A000984.
%K A378026 nonn,walk
%O A378026 0,2
%A A378026 _Markus Kuba_, Nov 14 2024
%E A378026 More terms from _Vaclav Kotesovec_, Nov 15 2024