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

0,2

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.

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

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 *)

Crossrefs

Cf. A000108, A060941, A322634.

Sequence in context: A349721 A252710 A065923 * A365652 A094476 A304637

Adjacent sequences: A293943 A293944 A293945 * A293947 A293948 A293949

Keyword

nonn,easy

Author

N. J. A. Sloane, Oct 24 2017

Extensions

More terms from Robert Israel, Oct 24 2017

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

Status

approved