

A190172


Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n having k UHD's; here U=(1,1), H=(1,0), and D=(1,1).


2



1, 1, 1, 1, 1, 2, 2, 4, 4, 8, 8, 1, 16, 18, 3, 33, 40, 9, 69, 90, 25, 1, 146, 204, 69, 4, 312, 467, 183, 16, 673, 1074, 479, 56, 1, 1463, 2481, 1239, 185, 5, 3202, 5752, 3180, 576, 25, 7050, 13378, 8104, 1734, 105, 1, 15605, 31196, 20544, 5076, 405, 6, 34705, 72912, 51852, 14546, 1451, 36
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,6


COMMENTS

Number of entries in row n is 1+floor(n/3).
Sum of entries in row n = A004148 (the RNA secondary structure numbers).
T(n,0)=A004149(n).
Sum(k*T(n,k),k>=0)=A110236(n2) (n>=3).


LINKS

Table of n, a(n) for n=0..62.


FORMULA

G.f. G=G(t,z) satisfies the equation G = 1 + zG + z^2*G(G1z+tz).


EXAMPLE

T(5,1)=4 because we have HHUHD, HUHDH, UHDH, and UUHDD.
Triangle starts:
1;
1;
1;
1,1;
2,2;
4,4;
8,8,1;
16,18,3;


MAPLE

eq := G = 1+z*G+z^2*G*(G1z+t*z): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 25)): for n from 0 to 17 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 17 do seq(coeff(P[n], t, k), k = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form


CROSSREFS

Cf. A004148, A004149, A110236
Sequence in context: A279405 A100835 A120541 * A287293 A059867 A046971
Adjacent sequences: A190169 A190170 A190171 * A190173 A190174 A190175


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, May 06 2011


STATUS

approved



