0,7

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.

M. S. Waterman, Home Page (contains copies of his papers)

M. Vauchassade de Chaumont and G. Viennot, Polynomes orthogonaux at problemes d'enumeration en biologie moleculaire, Sem. Loth. Comb. B08l (1984) 79-86.

T(0, 0)=1; T(n, k)=binomial(n-k, k)*binomial(n-k, k+1)/(n-k) for 2k<=n-1. G.f.=g=(1-z+tz^2-sqrt(1-2z+z^2-2tz^2-2tz^3+t^2*z^4))/(2tz^2), solution of g=1+zg+tz^2*g(g-1). G.f.=1+r(tz, z), where r(t, z) is the Narayana function defined by r=z(1+r)(1+tr). Column g.f.'s are 1/(1-z) for column 0 and z^(k+1)*N_k(z)/(1-z)^(2k+1) for columns k=1, 2, ..., where N_k(z)=(1/k)sum(binomial(k, j)*binomial(k, j-1)*z^(j-1), j=1..k) are the Narayana polynomials.

G.f. g(z, t) = Sum_{n, k} T(n, k)z^n*t^k = 1/(1 - z + z^2*t(1-g(z, t))). - Michael Somos Sep 08 2005

Given g.f. g(z, t) then G=z*g(z, t) series reversion in z is -G(-z, t). - Michael Somos Sep 08 2005

Given g.f. g(z, t) then G=z*g(z, t) satisfies G = z + z*G/(1-t*z*G). - Michael Somos Sep 08 2005

T(4,1)=3 because we have UHDH, HUHD and UHHD, where U=(1,1), D=(1,-1), H=(1,0).

1; 1; 1; 1,1; 1,3; 1,6,1; 1,10,6; 1,15,20,1; 1,21,50,10; 1,28,105,50,1;

(PARI) {T(n, k)=local(A); if(n<1, k==0, n--; A=1+O(x); for(i=1, (n+1)\2, A = 1/(1/(1+x*x*y*A)-x)); polcoeff(polcoeff(A, n), k))} /* Michael Somos Sep 08 2005 */

Row sums give A004148.

Sequence in context: A128489 A130541 A034839 this_sequence A158905 A098076 A010287

Adjacent sequences: A089729 A089730 A089731 this_sequence A089733 A089734 A089735

nonn,tabf

Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 07 2004