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.

1, 2, 19, 293, 5452, 112227, 2460954, 56356938, 1332055265, 32251721089, 795815587214, 19939653287183, 505943824579282, 12974266405435153, 335717028959470883, 8754495459668971998, 229836484204401559180, 6069875377376291350173, 161145418968823760038557

Offset: 0

Author

N. J. A. Sloane, Oct 24 2017

Keywords

• nonn
• easy

Links

• Robert Israel, Table of n, a(n) for n = 0..687 (corrected by Ray Chandler, Jan 19 2019)
• M. T. L. Bizley, 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, Journal of the Institute of Actuaries, Vol. 80, No. 1 (1954): 55-62.[Cached copy; Annotated copy of page 59]
• Bryan Ek, Lattice Walk Enumeration, arXiv:1803.10920 [math.CO], 2018.
• Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.

Crossrefs

Cf. A000108, A060941, A322634.

Programs

Maple

f:= proc(n) local U,x,y;
  U:= Array(1..3*n,0..2*n);
  U[3*n,2*n]:= 1:
  for x from 3*n to 1 by -1 do
    for y from ceil(2/3*x)-1 to 0 by -1 do
      if x+1 <= 3*n then U[x,y]:= U[x+1,y] fi;
      if y+1 < 2/3*x or x=3*n then U[x,y]:= U[x,y]+U[x,y+1] fi;
  od od:
  U[1,0];
end proc:
map(f, [$1..30]); # Robert Israel, Oct 24 2017

Mathematica

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]];
a[n_] := T[3n, 2n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 10 2018, after Danny Rorabaugh *)

Formula

a(n) = T(3n,2n) where T is the triangle from A294207. - Danny Rorabaugh, Oct 24 2017

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

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

Extensions

More terms from Robert Israel, Oct 24 2017

Offset changed and a(0) by Danny Rorabaugh, Oct 24 2017