A162981 Number of Dyck paths with no UUU's and no DDD's of semilength n and having k returns to the x-axis (1 <= k <= n; U=(1,1), D=(1,-1)).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 2, 4, 6, 4, 1, 4, 7, 10, 10, 5, 1, 8, 14, 18, 20, 15, 6, 1, 17, 29, 36, 39, 35, 21, 7, 1, 37, 62, 76, 80, 75, 56, 28, 8, 1, 82, 136, 165, 172, 161, 132, 84, 36, 9, 1, 185, 304, 366, 380, 355, 300, 217, 120, 45, 10, 1, 423, 690, 826, 855, 800, 684, 525

Offset: 1

Author

Emeric Deutsch, Oct 11 2009

Keywords

• nonn
• tabl

Comments

Sum of entries in row n = A004148(n+1) (the secondary structure numbers).

T(n,1) = A004148(n-2) (n>=2).

Sum_{k=1..n} k*T(n,k) = A162983(n).

Examples

			T(5,2)=4 because we have UD'UUDUDUDD', UUDD'UUDUDD', UUDUDD'UUDD', and UUDUDUDD'UD' (the return steps are marked).
Triangle starts:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,  1;
  2,  4,  6,  4,  1;
  4,  7, 10, 10,  5,  1;
		

Crossrefs

Cf. A004148, A162983.

Programs

Maple

g := ((1-z-z^2-sqrt(1-2*z-z^2-2*z^3+z^4))*1/2)/z^3: G := 1/(1-t*z-t*z^2-t*z^3*g)-1: Gser := simplify(series(G, z = 0, 16)): for n to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n to 12 do seq(coeff(P[n], t, j), j = 1 .. n) end do; # yields sequence in triangular form

Formula

G.f.: G(t,z) = 1/(1-tz-tz^2-tz^3*g) - 1, where g = 1 + zg + z^2*g + z^3*g^2.