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A089732
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Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n having k (1,1) steps (can be easily translated into RNA secondary structure terminology). Except for row 0, row n has ceil(n/2) entries.
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3
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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, 1, 36, 196, 175, 15, 1, 45, 336, 490, 105, 1, 1, 55, 540, 1176, 490, 21, 1, 66, 825, 2520, 1764, 196, 1, 1, 78, 1210, 4950, 5292, 1176, 28, 1, 91, 1716, 9075, 13860, 5292, 336, 1, 1, 105
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,7
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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.
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LINKS
| 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.
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FORMULA
| 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
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EXAMPLE
| 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;
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PROG
| (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 */
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CROSSREFS
| Row sums give A004148.
Sequence in context: A128489 A130541 A034839 * A158905 A098076 A171852
Adjacent sequences: A089729 A089730 A089731 * A089733 A089734 A089735
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 07 2004
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