OFFSET
1,2
COMMENTS
Row n has ceiling(n/2) terms. Row sums yield A110236.
LINKS
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 (1978), 261-272.
M. Vauchassade de Chaumont and G. Viennot, Polynômes orthogonaux et problèmes d'énumération en biologie moléculaire, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86.
FORMULA
G.f.: z*g^2/(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).
T(n,m) = (m+1)*Sum_{i=0..(n-1)/2-m}((binomial(2*m+2*i+2,i)*Sum_{k=0..n-2*m-2*i-1}(binomial(k,n-2*m-k-2*i-1)*binomial(2*m+k+2*i+1,k)*(-1)^(n-k-1)))/(m+i+1)). - Vladimir Kruchinin, Mar 07 2016
EXAMPLE
T(5,1)=10 because in the 8 (=A004148(5)) peakless Motzkin paths of length 5, namely HHHHH, U(H)DHH, U(HH)DH, U(HHH)D, HU(H)DH, HU(HH)D, HHU(H)D and UUHDD (where U=(1,1), H=(1,0) and D=(1,-1)), we have altogether 10 H steps at level 1 (shown between parentheses).
Triangle starts:
1;
2;
3, 1;
6, 4;
13, 10, 1;
MAPLE
g:=(1-z+z^2-sqrt(1-2*z-z^2-2*z^3+z^4))/2/z^2: G:=z*g^2/(1-t*z^2*g^2): Gser:=simplify(series(G, z=0, 20)): for n from 1 to 15 do P[n]:=coeff(Gser, z^n) od: for n from 1 to 15 do seq(coeff(t*P[n], t^k), k=1..ceil(n/2)) od;
PROG
(Maxima)
T(n, m):=(m+1)*sum((binomial(2*m+2*i+2, i)*sum(binomial(k, n-2*m-k-2*i-1)*binomial(2*m+k+2*i+1, k)*(-1)^(n-k-1), k, 0, n-2*m-2*i-1))/(m+i+1), i, 0, (n-1)/2-m); /* Vladimir Kruchinin, Mar 07 2016 */
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jul 17 2005
STATUS
approved