A191525 Triangle read by rows: T(n,k) is the number of left factors of Dyck paths of length n and having k hills; a hill is a (1,1)-step starting at level 0 and followed by a (1,-1)-step.

1, 1, 1, 1, 2, 1, 4, 1, 1, 7, 2, 1, 13, 5, 1, 1, 24, 8, 2, 1, 46, 16, 6, 1, 1, 86, 28, 9, 2, 1, 166, 58, 19, 7, 1, 1, 314, 103, 32, 10, 2, 1, 610, 211, 71, 22, 8, 1, 1, 1163, 382, 121, 36, 11, 2, 1, 2269, 781, 261, 85, 25, 9, 1, 1, 4352, 1432, 456, 140, 40, 12, 2, 1, 8518, 2920, 976, 316, 100, 28, 10, 1, 1

Offset: 0

Author

Emeric Deutsch, Jun 06 2011

Keywords

• nonn
• tabf

Comments

Row n contains 1+floor(n/2) entries.

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

T(n,0)=A191526(n).

Sum(k*T(n,k), k>=0) = A037952(n-1).

Examples

			T(5,1)=2 because we have (UD)UUD and (UD)UUU, where U=(1,1) and D=(1,-1) (the hills are shown between parentheses).
Triangle starts:
  1;
  1;
  1,1;
  2,1;
  4,1,1;
  7,2,1;
  13,5,1,1;
		

Crossrefs

Cf. A001405, A191526, A037952, A065600.

Programs

Maple

c := ((1-sqrt(1-4*z^2))*1/2)/z^2: G := 1/((1-z*c)*(1-z^4*c^2-t*z^2)): Gser := simplify(series(G, z = 0, 19)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

Formula

G.f.: G(t,z)=1/((1-z*c)*(1-z^4*c^2-t*z^2)), where c=(1-sqrt(1-4*z^2))/(2*z^2) is the Catalan function with argument z^2.