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
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