A191781 Triangle read by rows: T(n,k) is the number of length-n left factors of Dyck paths having length of the first ascent equal to k (1 <= k <= n).

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 3, 2, 1, 1, 6, 6, 4, 2, 1, 1, 10, 10, 7, 4, 2, 1, 1, 20, 20, 14, 8, 4, 2, 1, 1, 35, 35, 25, 15, 8, 4, 2, 1, 1, 70, 70, 50, 30, 16, 8, 4, 2, 1, 1, 126, 126, 91, 56, 31, 16, 8, 4, 2, 1, 1, 252, 252, 182, 112, 62, 32, 16, 8, 4, 2, 1, 1, 462, 462, 336, 210, 119, 63, 32, 16, 8, 4, 2, 1, 1

Offset: 1

Author

Emeric Deutsch, Jun 18 2011

Keywords

• nonn
• tabl

Comments

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

T(n,1) = A001405(n-2) (n>=1).

T(n,2) = A001405(n-2) (n>=2).

Sum_{k=1..n} k*T(n,k) = A191782(n).

Examples

			T(5,2)=3 because we have UUDDU, UUDUD, and UUDUU, where U=(1,1) and D=(1,-1).
Triangle starts:
  1;
  1, 1;
  1, 1, 1;
  2, 2, 1, 1;
  3, 3, 2, 1, 1;
  6, 6, 4, 2, 1, 1;
		

Crossrefs

Cf. A001405, A191782.

Programs

Maple

c := ((1-sqrt(1-4*z^2))*1/2)/z^2: G := (1-z*c+t*z^3*c^2)/((1-z*c)*(1-t*z)*(1-t*z^2*c)): Gser := simplify(series(G, z = 0, 17)): for n to 13 do P[n] := sort(coeff(Gser, z, n)) end do: for n to 13 do seq(coeff(P[n], t, k), k = 1 .. n) end do; # yields sequence in triangular form

Formula

G.f.: G(t,z) = (1 - z*c + t*z^3*c^2)/((1 - z*c)*(1 - t*z)*(1 - t*z^2)), where c = (1-sqrt(1 - 4*z^2))/(2*z^2).