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A110238 Triangle read by rows: T(n,k) (0<=k<=ceil(n/2)-2) is the number of (1,1) steps starting at level k in all peakless Motzkin paths of length n (can be easily translated into RNA secondary structure terminology). 1
1, 3, 7, 1, 17, 5, 41, 16, 1, 98, 46, 7, 235, 127, 29, 1, 565, 340, 99, 9, 1362, 893, 310, 46, 1, 3294, 2318, 921, 184, 11, 7992, 5968, 2640, 650, 67, 1, 19450, 15279, 7382, 2131, 309, 13, 47475, 38965, 20274, 6641, 1223, 92, 1, 116204, 99101, 54935, 19958, 4404 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,2

COMMENTS

Row n has ceil(n/2)-1 terms. Row sums yield A110239

REFERENCES

W. R. Schmitt and M. S. Waterman, Linear trees and RNA secondary structure, Discrete Appl. Math., 51, 317-323, 1994.

P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26, 1979, 261-272.

M. Vauchassade de Chaumont and G. Viennot, Polynomes orthogonaux et problemes d'enumeration en biologie moleculaire, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86.

LINKS

Table of n, a(n) for n=3..56.

FORMULA

G.f.=z^2*g^2*(g-1)/(1-tz^2*g^2), where g=1+zg+z^2*g(g-1)=[1-z+z^2-sqrt(1-2z-z^2-2z^3+z^4)]/(2z^2) is the g.f. of the RNA secondary structure numbers (A004148).

EXAMPLE

T(5,1)=1 because in the 8 (=A004148(5)) peakless Motzkin paths of length 5, namely HHHHH, UHDHH, UHHDH, UHHHD, HUHDH, HUHHD, HHUHD and U(U)HDD (where U=(1,1), H=(1,0) and D=(1,-1)), we have altogether 1 U step starting at level 1 (shown between parentheses).

Triangle starts:

1;

3;

7,1;

17,5;

41,16,1;

98,46,7;

MAPLE

g:=(1-z+z^2-sqrt(1-2*z-z^2-2*z^3+z^4))/2/z^2: G:=z^2*g^2*(g-1)/(1-t*z^2*g^2): Gser:=simplify(series(G, z=0, 21)): for n from 1 to 17 do P[n]:=coeff(Gser, z^n) od: for n from 3 to 17 do seq(coeff(t*P[n], t^k), k=1..ceil(n/2)-1) od;

CROSSREFS

Cf. A004148, A110239, A110237.

Sequence in context: A210197 A324715 A316665 * A077505 A247382 A031436

Adjacent sequences:  A110235 A110236 A110237 * A110239 A110240 A110241

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Jul 17 2005

STATUS

approved

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Last modified July 17 02:51 EDT 2019. Contains 325092 sequences. (Running on oeis4.)