A162981 - OEIS

%I #7 Jul 21 2017 02:47:24

%S 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,

%T 36,39,35,21,7,1,37,62,76,80,75,56,28,8,1,82,136,165,172,161,132,84,

%U 36,9,1,185,304,366,380,355,300,217,120,45,10,1,423,690,826,855,800,684,525

%N 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)).

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

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

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

%F 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.

%e T(5,2)=4 because we have UD'UUDUDUDD', UUDD'UUDUDD', UUDUDD'UUDD', and UUDUDUDD'UD' (the return steps are marked).

%e Triangle starts:

%e   1;

%e   1,  1;

%e   1,  2,  1;

%e   1,  3,  3,  1;

%e   2,  4,  6,  4,  1;

%e   4,  7, 10, 10,  5,  1;

%p 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

%Y Cf. A004148, A162983.

%K nonn,tabl

%O 1,5

%A _Emeric Deutsch_, Oct 11 2009