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A098086 Triangle read by rows: T(n,k) is the number of peakless Motzkin paths in which the k-th step is the leftmost (1,0)-step (can be easily expressed using RNA secondary structure terminology). 0
1, 1, 1, 1, 2, 2, 4, 3, 1, 8, 6, 3, 17, 13, 6, 1, 37, 28, 13, 4, 82, 62, 30, 10, 1, 185, 140, 69, 24, 5, 423, 320, 160, 59, 15, 1, 978, 740, 375, 144, 40, 6, 2283, 1728, 885, 350, 105, 21, 1, 5373, 4068, 2102, 852, 271, 62, 7, 12735, 9645, 5022, 2077, 690, 174, 28, 1, 30372 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Row sums yield the RNA secondary structure numbers (A004148).

Row n has ceiling(n/2) terms.

LINKS

Table of n, a(n) for n=1..65.

I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.

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, Polynômes orthogonaux et problèmes d'énumération en biologie moléculaire, Sem. Loth. Comb. B08l (1984) 79-86.

FORMULA

T(n,k) = k*Sum_{j=ceiling((n-2k+2)/2)..n-2k+1} (1/j)*binomial(j, n-2k+1-j)*binomial(j+k-1, n-k+1-j) if 2k < n+1 and T(n,k)=1 if 2k = n+1.

G.f.: t*z*g/(1-t*z^2*g), where g = (1 - z + z^2 - sqrt(1 - 2*z - z^2 - 2*z^3 + z^4))/(2z^2) is the g.f. for the RNA secondary structure numbers (A004148).

EXAMPLE

Triangle starts:

   1;

   1;

   1,  1;

   2,  2;

   4,  3,  1;

   8,  6,  3;

  17, 13,  6,  1;

T(5,2)=3 because we have UHDHH, UHHDH and UHHHD, where U=(1,1), H=(1,0) and D=(1,-1); these are the only peakless Motzkin paths of length 5 in which the second step is the first occurrence of H.

MAPLE

T:=proc(n, k) if 2*k-1=n then 1 else k*sum(binomial(j, n-2*k+1-j)*binomial(j+k-1, n-k+1-j)/j, j=ceil((n-2*k+2)/2)..n-2*k+1) fi end: seq(seq(T(n, k), k=1..ceil(n/2)), n=0..16); # yields the sequence in linear form T:=proc(n, k) if 2*k-1=n then 1 else k*sum(binomial(j, n-2*k+1-j)*binomial(j+k-1, n-k+1-j)/j, j=ceil((n-2*k+2)/2)..n-2*k+1) fi end: matrix(10, 10, T); # yields the sequence in triangular form

CROSSREFS

Cf. A004148.

Sequence in context: A349474 A105114 A166284 * A332887 A306323 A317015

Adjacent sequences:  A098083 A098084 A098085 * A098087 A098088 A098089

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Sep 13 2004

STATUS

approved

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Last modified November 29 12:24 EST 2021. Contains 349416 sequences. (Running on oeis4.)