OFFSET
0,12
REFERENCES
A. Nkwanta, Lattice paths and RNA secondary structures, DIMACS Series in Discrete Math. and Theoretical Computer Science, 34, 1997, 137-147.
LINKS
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. [Formerly: Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, p. 79-86.]
M. S. Waterman, Home Page (contains copies of his papers)
FORMULA
G.f.= g/(1+zg-tzg), where g := (1-z+z^2-sqrt(1-2z-z^2-2z^3+z^4))/(2z^2) is the g.f. of A004148.
T(n,m) = Sum_{j=0..n-m}((m+j+1)*binomial(m+j,j)*Sum_{i=0..(n-j+1)/2 }((binomial(m+j+2*i+1,i)*Sum_{k=0..n-m-j-2*i}(binomial(k,n-m-k-j-2*i)*binomial(m+k+j+2*i,k)*(-1)^(n-m-k)))/(m+j+2*i+1))). - Vladimir Kruchinin, Mar 07 2016
EXAMPLE
T(5,2)=3 because we have H'H'UHD, H'UHDH' and UHDH'H', where U=(1,1), D=(1,-1), H=(1,0) and H' indicates an H step at level zero.
1; 0,1; 0,0,1; 1,0,0,1; 1,2,0,0,1; 2,2,3,0,0,1; 5,4,3,4,0,0,1; 10,11,6,4,5,0,0,1; 22,22,18,8,5,6,0,0,1;
PROG
(Maxima)
T(n, m):=sum((m+j+1)*binomial(m+j, j)*sum((binomial(m+j+2*i+1, i)*sum(binomial(k, n-m-k-j-2*i)*binomial(m+k+j+2*i, k)*(-1)^(n-m-k), k, 0, n-m-j-2*i))/(m+j+2*i+1), i, 0, (n-j+1)/2), j, 0, n-m); /* Vladimir Kruchinin, Mar 07 2016 */
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Jan 07 2004
STATUS
approved