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A293946 a(n) = number of lattice paths from (0,0) to (3n,2n) which lie wholly below the line 3y=2x, only touching at the endpoints.

%I A293946 #55 Sep 16 2021 11:01:52
%S A293946 1,2,19,293,5452,112227,2460954,56356938,1332055265,32251721089,
%T A293946 795815587214,19939653287183,505943824579282,12974266405435153,
%U A293946 335717028959470883,8754495459668971998,229836484204401559180,6069875377376291350173,161145418968823760038557
%N A293946 a(n) = number of lattice paths from (0,0) to (3n,2n) which lie wholly below the line 3y=2x, only touching at the endpoints.
%H A293946 Robert Israel, <a href="/A293946/b293946.txt">Table of n, a(n) for n = 0..687</a> (corrected by Ray Chandler, Jan 19 2019)
%H A293946 M. T. L. Bizley, <a href="http://bergeron.math.uqam.ca/wp-content/uploads/2014/09/Bizley.pdf">Derivation of a new formula for the number of minimal lattice paths from (0, 0) to (km, kn) having just t contacts with the line my = nx and having no points above this line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line</a>, Journal of the Institute of Actuaries, Vol. 80, No. 1 (1954): 55-62.[<a href="/A060941/a060941.pdf">Cached copy</a>; <a href="/A060941/a060941.png">Annotated copy of page 59</a>]
%H A293946 Bryan Ek, <a href="https://arxiv.org/abs/1803.10920">Lattice Walk Enumeration</a>, arXiv:1803.10920 [math.CO], 2018.
%H A293946 Bryan Ek, <a href="https://arxiv.org/abs/1804.05933">Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics</a>, arXiv:1804.05933 [math.CO], 2018.
%F A293946 a(n) = T(3n,2n) where T is the triangle from A294207. - _Danny Rorabaugh_, Oct 24 2017
%F A293946 G.f. A(z) satisfies A^10-19*A^9+162*A^8-816*A^7+2688*A^6+(-2*z-6048)*A^5+(19*z+9408)*A^4+(-73*z-9984)*A^3+(142*z+6912)*A^2+(-140*z-2816)*A+z^2+56*z+512=0 (Proven). - _Bryan T. Ek_, Oct 30 2017
%F A293946 a(n) ~ (2 + 10^(1/3)) * 5^(5*n - 3/2) / (sqrt(Pi) * n^(3/2) * 2^(2*n + 1) * 3^(3*n + 1/2)). - _Vaclav Kotesovec_, Sep 16 2021
%p A293946 f:= proc(n) local U,x,y;
%p A293946   U:= Array(1..3*n,0..2*n);
%p A293946   U[3*n,2*n]:= 1:
%p A293946   for x from 3*n to 1 by -1 do
%p A293946     for y from ceil(2/3*x)-1 to 0 by -1 do
%p A293946       if x+1 <= 3*n then U[x,y]:= U[x+1,y] fi;
%p A293946       if y+1 < 2/3*x or x=3*n then U[x,y]:= U[x,y]+U[x,y+1] fi;
%p A293946   od od:
%p A293946   U[1,0];
%p A293946 end proc:
%p A293946 map(f, [$1..30]); # _Robert Israel_, Oct 24 2017
%t A293946 T[_, 0] = 1; T[n_, k_] := T[n, k] = Which[0 < k < 2(n-1)/3, T[n-1, k] + T[n, k-1], 2(n-1) <= 3k <= 2n, T[n, k-1]];
%t A293946 a[n_] := T[3n, 2n];
%t A293946 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jul 10 2018, after _Danny Rorabaugh_ *)
%Y A293946 Cf. A000108, A060941, A322634.
%K A293946 nonn,easy
%O A293946 0,2
%A A293946 _N. J. A. Sloane_, Oct 24 2017
%E A293946 More terms from _Robert Israel_, Oct 24 2017
%E A293946 Offset changed and a(0) by _Danny Rorabaugh_, Oct 24 2017