A191795 Triangle read by rows: T(n,k) is the number of length n left factors of Dyck paths having k DUU's, where U=(1,1) and D=(1,-1).

1, 1, 2, 3, 5, 1, 7, 3, 11, 9, 15, 19, 1, 23, 42, 5, 31, 77, 18, 47, 150, 54, 1, 63, 255, 137, 7, 95, 464, 333, 32, 127, 753, 720, 115, 1, 191, 1314, 1558, 360, 9, 255, 2067, 3067, 996, 50, 383, 3508, 6167, 2597, 214, 1, 511, 5397, 11410, 6207, 774, 11, 767, 8982, 21820, 14485, 2494, 72

Offset: 0

Author

Emeric Deutsch, Jun 18 2011

Keywords

• nonn
• tabf

Comments

Row n >= 1 contains ceiling(n/3) entries.

Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).

T(n,0) = A052955(n-1).

Sum_{k>=0} k*T(n,k) = A191796(n).

Examples

			T(5,1)=3 because we have U(DUU)D, U(DUU)U, and UU(DUU), where U=(1,1) and D=(1,-1) (the DUU's are shown between parentheses).
Triangle starts:
   1;
   1;
   2;
   3;
   5,  1;
   7,  3;
  11,  9;
  15, 19, 1;
  23, 42, 5;
		

Crossrefs

Cf. A001405, A052955, A191796.

Programs

Maple

eq := t*z^2*C^2-(1-2*z^2+2*t*z^2)*C+1-z^2+t*z^2 = 0: C := RootOf(eq, C): G := 1-(1-C-z*C)/(1-z+t*z-t*z*C): Gser := simplify(series(G, z = 0, 23)): for n from 0 to 18 do P[n] := sort(coeff(Gser, z, n)) end do: 1; for n to 18 do seq(coeff(P[n], t, k), k = 0 .. ceil((1/3)*n)-1) end do; # yields sequence in triangular form

Formula

G.f.: G(t,z) = 1 - (1-C-z*C)/(1-z+t*z-t*z*C), where C=C(t,z) is given by t*z^2*C^2 - (1-2*z^2+2*t*z^2)*C + 1-z^2+t*z^2 = 0.