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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.
6
1, 2, 19, 293, 5452, 112227, 2460954, 56356938, 1332055265, 32251721089, 795815587214, 19939653287183, 505943824579282, 12974266405435153, 335717028959470883, 8754495459668971998, 229836484204401559180, 6069875377376291350173, 161145418968823760038557
OFFSET
0,2
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