A127167 T(3n,2n), where T is the array in A047110.

1, 5, 73, 1348, 27811, 612728, 14103464, 334974405, 8146520511, 201822398131, 5074951075766, 129185072614240, 3322359273912432, 86191998671455630, 2252923797781463037, 59273686760263160137, 1568440076774389559527, 41713725234702452284249

Offset: 0

Author

Clark Kimberling, Jan 06 2007

Keywords

• nonn

Examples

			T(h,k) counts lattice paths restricted as in A047110 by the line y=2x/3, so that the numbers T(3n,2n) are of interest.
T(3,2) counts these 5 paths: RRRUU, RRURU, URRRU, URURR, UURRR.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A047110.

Programs

Maple

T:= proc(h, k) option remember;
      `if`([h, k]=[0, 0], 1, `if`(h<0 or k<0, 0, T(h-1, k)+
      `if`(3*k>2*h and 3*(k-1)<2*h, 0, T(h, k-1))))
    end:
a:= n-> T(3*n, 2*n):
seq (a(n), n=0..30); # Alois P. Heinz, Apr 04 2012
series(RootOf((161051*x^2+1088*x-64)*A^10 + (4600*x-320)*A^9 + (7276*x-624)*A^8 + (7830*x-512)*A^7 + (6189*x+68)*A^6 + (2177*x+540)*A^5 +535*A^4 +278*A^3 +84*A^2 +14*A +1, A), x=0, 30); # Mark van Hoeij, May 01 2013

Mathematica

T[h_, k_] := T[h, k] = If[{h, k} == {0, 0}, 1, If[h < 0 || k < 0, 0, T[h-1, k] + If[3k > 2h && 3(k-1) < 2h, 0, T[h, k-1]]]];
a[n_] := T[3n, 2n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 21 2025, after Alois P. Heinz *)

Formula

G.f.: A(x) where (161051*x^2 +1088*x -64) * A(x)^10 + (4600*x -320) * A(x)^9 + (7276*x -624) * A(x)^8 + (7830*x -512) * A(x)^7 + (6189*x +68) * A(x)^6 + (2177*x + 540) * A(x)^5 + 535*A(x)^4 + 278*A(x)^3 + 84*A(x)^2 + 14*A(x) + 1 = 0. - Mark van Hoeij, May 01 2013

Extensions

More terms from Alois P. Heinz, Apr 04 2012